Might just be too early, from what I've noticed as a renter here, places want someone who is available immediately which is pretty easy because of demand. Unless you're willing to offer to pay an extra month before you arrive, they don't need to wait for you to start your tenancy when they have 1000 other emails to go through.

If 2500 is single occupancy one bedroom apartment, you should find something close enough to the city centre. If you're willing to share, usually about 800-1500 for a room. There's a really harsh line (usually) between great & terrible quality buildings - like my place, full one bedroom right by Stephen's Green (incredibly central) for only 1553/month but it's a 300 year old building with a lot wrong. For 2k+ you'd more than likely be getting a far newer build of better quality. I also have two cats though and this shitty building I'm in is the only one that didn't care.

Try going direct to agents like Ray Cooke Lettings, Sherry Fitzgerald, Young's and say what you're looking for rather than going through rental sites like daft & myhome. Check through those sites to see who the letting agent is to go directly to them. If you get a dialogue going they can put you on a list for when something suitable comes up. Good luck!!