How did the suffering of Gautama Buddha arise or co-arise? As per my understanding during his lifetime, Gautama Buddha, had attained Nibbana with residue. And after death he attained Parinibbana (or Nibbana with no residue).

Nibbana with residue is defined as following :


  What, bhikkhus, is the Nibbana-element with residue left? Here a
  bhikkhu is an arahant, one whose taints are destroyed, the holy life
  fulfilled, who has done what had to be done, laid down the burden,
  attained the goal, destroyed the fetters of being, completely released
  through final knowledge. However, his five sense faculties remain
  unimpaired, by which he still experiences what is agreeable and
  disagreeable and feels pleasure and pain. It is the extinction of
  attachment, hate, and delusion in him that is called the
  Nibbana-element with residue left.


During his last days Gautama Buddha felt sharp pain and illness which he suppressed. In other words Gautama Buddha suffered because he was yet to reach his final Nibbana (Nibbana with no residue).


  And soon after the Blessed One had eaten the meal provided by Cunda
  the metalworker, a dire sickness fell upon him, even dysentery, and he
  suffered sharp and deadly pains. But the Blessed One endured them
  mindfully, clearly comprehending and unperturbed.


We describe all sufferings using dependent co-arising.But dependent co-arising of suffering of Gautama Buddha is only possible if Gautama Buddha had failed to achieve the complete cessation of ignorance. Because if he had completely destroyed the ignorance at the very beginning then suffering would have ceased automatically.  


  From the remainderless fading &amp; cessation of that very ignorance comes
  the cessation of fabrications.  From the cessation of fabrications
  comes the cessation of consciousness.  From the cessation of
  consciousness comes the cessation of name-&amp;-form.  From the cessation
  of name-&amp;-form comes the cessation of the six sense media.  From the
  cessation of the six sense media comes the cessation of contact.  From
  the cessation of contact comes the cessation of feeling.  From the
  cessation of feeling comes the cessation of craving.  From the
  cessation of craving comes the cessation of clinging/sustenance.  From
  the cessation of clinging/sustenance comes the cessation of becoming. 
  From the cessation of becoming comes the cessation of birth.  From the
  cessation of birth, then aging &amp; death, sorrow, lamentation, pain,
  distress, &amp; despair all cease.  Such is the cessation of this entire
  mass of stress &amp; suffering


My question is: How did the suffering of Gautama Buddha(no matter how small) arise or co-arise?

My guess is that with incomplete cessation of ignorance came incomplete cessation of fabrications. From incomplete cessation of fabrications came incomplete cessation of consciousness. From incomplete cessation of consciousness came incomplete cessation of name-form. From incomplete cessation of name-form came incomplete cessation of six sense media. From incomplete cessation of six sense media came incomplete cessation of contact. From incomplete cessation of contact came incomplete cessation of feeling. From incomplete cessation of feeling came incomplete craving. From incomplete cessation of craving came incomplete cessation of clinging/sustenance.  From incomplete cessation of clinging/sustenance came incomplete cessation of becoming. From incomplete cessation of becoming came incomplete cessation of birth. From incomplete cessation of birth came incomplete cessation of aging ,death ,sorrow,pain , distress.

I am not saying that Buddha was ignorant but that there is a possibility that there are levels of ignorance which one needs to transcend like the levels of jhanas.
"What, bhikkhus, is the Nibbana-element with residue left? Here a
  bhikkhu is an arahant, one whose taints are destroyed, the holy life
  fulfilled, who has done what had to be done, laid down the burden,
  attained the goal, destroyed the fetters of being, completely released
  through final knowledge. However, his five sense faculties remain
  unimpaired, by which he still experiences what is agreeable and
  disagreeable and feels pleasure and pain. It is the extinction of
  attachment, hate, and delusion in him that is called the
  Nibbana-element with residue left.


According to Itivuttaka 44 (above), an Arahant (including the Buddha) can still feel pleasure and pain.


  And soon after the Blessed One had eaten the meal provided by Cunda
  the metalworker, a dire sickness fell upon him, even dysentery, and he
  suffered sharp and deadly pains. But the Blessed One endured them
  mindfully, clearly comprehending and unperturbed.


The suffering had indeed ended for the Buddha. That was why he was unperturbed by physical ailments. He feels pain but does not suffer from it.

In the Sivaka Sutta, the Buddha explains that not everything one feels is due to past karma becoming ripe:


  "There are cases where some feelings arise based on
  bile. You yourself should know how some feelings arise based on bile.
  Even the world is agreed on how some feelings arise based on bile. So
  any brahmans &amp; contemplatives who are of the doctrine &amp; view that
  whatever an individual feels — pleasure, pain,
  neither-pleasure-nor-pain — is entirely caused by what was done before
  — slip past what they themselves know, slip past what is agreed on by
  the world. Therefore I say that those brahmans &amp; contemplatives are
  wrong."
  
  "There are cases where some feelings arise based on phlegm... based on
  internal winds... based on a combination of bodily humors... from the
  change of the seasons... from uneven care of the body... from harsh
  treatment... from the result of kamma. You yourself should know how
  some feelings arise from the result of kamma. Even the world is agreed
  on how some feelings arise from the result of kamma. So any brahmans
  &amp; contemplatives who are of the doctrine &amp; view that whatever an
  individual feels — pleasure, pain, neither pleasure-nor-pain — is
  entirely caused by what was done before — slip past what they
  themselves know, slip past what is agreed on by the world. Therefore I
  say that those brahmans &amp; contemplatives are wrong."


In the story of Ven. Chakkhupala in Dhammapada 1, we find that it is possible for an Arahant to be blind due to past karma becoming ripe prior to Nibbana.

But since he was already blind, he does not suddenly become cured and starts to have vision because of Nibbana.


  On one occasion, Thera Cakkhupala came to pay homage to the Buddha at
  the Jetavana monastery. One night, while pacing up and down in
  meditation, the thera accidentally stepped on some insects. In the
  morning, some bhikkhus visiting the thera found the dead insects. They
  thought ill of the thera and reported the matter to the Buddha. The
  Buddha asked them whether they had seen the thera killing the insects.
  When they answered in the negative, the Buddha said, "Just as you had
  not seen him killing, so also he had not seen those living insects.
  Besides, as the thera had already attained arahatship he could have no
  intention of killing and so was quite innocent." On being asked why
  Cakkhupala was blind although he was an arahat, the Buddha told the
  following story:
  
  Cakkhupala was a physician in one of his past existences. Once, he had
  deliberately made a woman patient blind. That woman had promised him
  to become his slave, together with her children, if her eyes were
  completely cured. Fearing that she and her children would have to
  become slaves, she lied to the physician. She told him that her eyes
  were getting worse when, in fact, they were perfectly cured. The
  physician knew she was deceiving him, so in revenge, he gave her
  another ointment, which made her totally blind. As a result of this
  evil deed the physician lost his eyesight many times in his later
  existences.


So, we find that no new karma becomes ripe after Nibbana. But it is possible that the Arahant could feel pain due to other reasons.

At the same time, any chronic conditions caused by previous karma becoming ripe prior to Nibbana, do not suddenly disappear due to Nibbana.

But in all cases, the Arahant is unperturbed. He feels pain but does not suffer from it. That shows that the Arahant has risen above suffering.

In this resource about dependent origination, Mahasi Sayadaw says:


  One who watches these present resultant processes effectively does not
  have craving that is rooted in feeling and so will put an end to the
  cycle of existence completely. In other words, the meditator watches
  every mental and physical phenomenon that occurs at the six
  sense-doors clearly in terms of its three characteristics. Through
  this effective practice of mindfulness, the meditator gains insight
  into the nature of the sense-objects and overcomes attachment to them
  there and then (tadaṅga), that is he or she overcomes it by opposing
  it with the knowledge that undercuts it. The cessation of attachment
  rules out the arising of the other phenomena, e.g. clinging, the
  process of becoming, birth, etc. After the this cessation through
  insight the meditator overcomes the latent attachment completely
  through cutting-off (samuccheda) when he or she attains the knowledge
  of the Noble Path. At this moment the other phenomena, e.g. clinging
  etc., also become totally extinct. 
  
  There is no teaching that says: “With the extinction of feeling, craving ceases to exist.” This is not surprising for even Arahants do
  not have any control over the feelings that arise from contact with
  the six senses.