Since we only see Omega in a post-Time Compression world, I would say, yes, he can survive it.
Ultimecia had intended to compress time to a point where only she could survive. Her spell failed, though we aren't sure why. Laguna proposed that the friendship and belief in each other would let everyone survive past Time Compression. If so, she completed her spell and it simply wasn't as powerful as she would like to have us believe. Or perhaps Ellone pulling Rinoa out of the past early disrupted the spell and prevented a full compression of time (although, since she supposedly left Ultimecia back there, I don't see why).
Either way, the result is the same: Ulti played her big magic, and it failed. Either she wasn't powerful enough to do what she needed to, or she didn't have enough support to pull it off. Either way, Omega survived her attempt, and is still around, not under her control, and far more powerful than she is.
Why he's in her castle, I'm not sure. But it seems fairly clear that he withstood her attempts to overcome him.
Although, perhaps Omega is a result of Time Compression, rather than a survivor of it. A fusion of monsters past and present, combined into an ultimate beast.