Just three points to note.

1. AirAsia operates from Terminal 1, while UA operates from Terminal 3. To get from one to the other you can take the skytrain (there's one that runs airside, another that runs groundside). This is reiterating what the other poster said.
2. If you change your mind on the baggage and end up checking bags with AirAsia, note you will need to go through Sing Immigration and Customs to retrieve your bag and then check in with UA ground side - AirAsia does not interline.
3. If you don't have any check in bags and therefore stay airside throughout, note that the transfer desk you need for UA is in Terminal 3, not Terminal 1. The transit desk is on the way to the gates that UA typically uses from UA's contract lounge.

All this is possible because at Changi Airport, there's no separation between inbound and outbound passengers airside.