There are no serviced appartments or other residential options near the zoo, which is in the jungle-covered centre of Singapore island. The nearest towns are Ang Mo Kio to the East and Woodlands to the North. Both have bus connections to the zoo, some foreign population (especially Woodlands due to the American School), but no Serviced Apartments
as far as I know.
You need to search nearer to the city!
A serviced 1-room studio will cost from S$2500/3000 up (sky's the limit) per month. You can get a basic hotel for a similar price. (Again, there are no hotels in Ang Mo Kio, Woodlands, or anywhere near the zoo.)
As far as I know, your cat will have to go into quarantine for a month, so that takes care of the first month. After that, you'd have to enquire with the apartment management. Most will say No.