We're getting new downstairs neighbours soon who have a cat. I think it's perfectly fine to bring them a bottle of wine as a house warming gift and tell them that if they're ever away we'd be happy to feed and play with their cat which I would assume would result in them saying they'd do the same for us. DP thinks it's a bit forward but I would be over the moon if I moved into a new place and my neighbours made that offer.
I think DP's issue might be that we're going to be away for a couple of days over Christmas and don't want to put Rice in a cattery so it would be nice if our neighbours could feed her and play with her while we're away if they're planning to be here. Equally though it could be that they're planning to be away longer than us so we'd be happy to feed their cat. I'm also thinking that maybe we could get a cat sitter for when we're away and negotiate a cheaper rate for looking after two cats in adjacent properties since travel time is built into the cost.