While the Original Gilgamesh is a villain in the original Fate/Stay Night, this version is more of an Anti Hero and helps Hakuno Kishinami in his/her journey.
Gilgamesh is unable to exist on the Near Side of the Moon Cell due to his strength, so he is only able to exist within the Far Side. He simply passes away the time sleeping due to nothing being able to exist there, but his physical body awakens with BB's restructuring of the Far Side
Gilgamesh's personality is heavily influenced by the era into which he is summoned. The world, his property, of Fate/stay night was sullied by the consumption society of the early 2000s, so he was in a worse mood overall. He has taken a liking to the virtual world of 2032, so his personality is more stable and closer to how it was during life. Arco Wada says he reminds her of Gilgamesh in Fate/hollow ataraxia. He still displays his normal self-centered traits, but Nasu wished to show that there is more to his character than that. He has no particular interest in the events within SE.RA.PH, and decides to simply do as he pleases. This later changes when he personally declared that the fight against BB has also become his battle.
Gilgamesh holds no romantic feelings for his Master, and he even goes as far to ask the female version "Who is the man you want to be held by?" for entertainment. He believes that there can be nothing else between a man and a woman. He restricts it to the Student Council members, but he will be annoyed if he is chosen instead. He takes it as a given that he is already the obvious choice, so it is useless in staving off boredom. He says it was unavoidable due to being exposed to his aura at such proximity and forgives her, but also completely rejects her, mentioning that she is four thousand years too early for him. He prefers "a noble maiden", a virgin", and "a flower that blooms in the plain" was his type during his childhood.