I have 1 piece of unrezzed ice protecting a server with CT upgrade face down. Runner approaches ice. Can I Rez ice and then Rez CT to pump ice before runner can break?

Yes.

Can I wait for them to pump strength of an ice breaker before using CT?

The player whose turn it is gets to do his paid abilities first, so once the runner starts using his icebreaker in step 3.1, you don't get to trigger any abilities until the runner is done breaking the subroutines.