Do cell phones and email exist in Gankutsuou? For the most part, Gankutsuou is set in Paris in the year 5053. Technological advancements shown in the series include:


Space travel and moon colonisation
Danglars' communication with Michelle, his assistant, over a wireless connection
Noirtier de Villefort's prosthetic arm
The pen inside Noirtier's arm, which auto-writes and which is used to access his memories
Lucien Debray's use of headsets to access government records through a VR application
The Count's pocketwatch, which is revealed to be a bugging device


With this taken into account, I was surprised to realise that the main characters never seem to use cell phones or email, even in situations such as:


Albert's failed attempted to see Eugénie after her engagement to Andrea Cavalcanti
Eugénie's thwarted attempt to see Albert after the Morcerf scandal breaks out
Eugénie's failed attempts to contact Franz while he was in Marseilles (although it is also possible that Franz intentionally made himself unreachable)


Do cell phones and email exist in the world of Gankutsuou? If they do, why do the characters not use them? Or are the technological advancements in Gankutsuou just constructed so that the original social structure  of the novel is not changed?
Gankutsuou's is an adaptation of The Count of Monte Cristo written by Alexandre Dumas and published in 1844. This is shown by the series' full English title: Gankutsuou: The Count of Monte Cristo.

The events of the original book take place in late 1830s France and as such the characters did not have access to technologies such as the telephone which would not be invented until the 1870s.

The futuristic setting of Gankutsuou is an adaptational choice and is very different setting to the original novel. This massive change in setting creates Adaptation-Induced Plot Holes such as this.

So your final question would be correct: Advancements such as cell phones are simply ignored in order for Gankutsuou to properly follow the plot of the original novel.