How important is patience in buddhism? I've heard that patience has a big place in Buddhism. It is especially very important for monks. I can think two ways of patience.

When we get angry, we have to have patience to avoid bad consequences.
Most of practises in buddhism don't give us a result immediately. So we have to have patience to see a result. (Delayed gratification)

What are the teachings of the Buddha about patience ? I can think of case 1. But I want to know about case 2.
If a person is unable to delay gratification (not only in Buddhist practises), does it means that the person has excessive clinging? (Is it a characteristic of that kind of person ?)
Khanti/patience is indeed a very important virtue to cultivate. It's listed in Sn 2.4 as among one of the greatest protection for a practitioner. Also refer to many other related suttas.

Patience, compliance,
seeing contemplatives,
discussing the Dhamma on timely occasions:
This is the highest protection. ~~ Snp 2.4 ~~