Andy's option is lowest cost, and should you want to upgrade you can always use the digital out from the Airport Epress into a standalone DAC. But assuming Andy is right about FLAC compatability (no reason to think he's not, and it does accord with Apple's lack of support for FLAC) then you can't play those files.
A dearer option would be to use something like the Arcam rDAC with its streaming dongle - not sure of the precise cost but it's around your budget. It's a great DAC (I have a wired one in an Ayre / Harbeth setup). I think, stress think, it can handle FLAC, but do please check if you go down this route.
What would I do? Start low cost and go for the Airport Express option, buying some software to convert the FLAC files into something it can handle. I use dbPoweramp, which converts FLAC to virtually any format you want. It's excellent but is PC only, but there are MAC equivalents, some of whch are free I think. You should be able to get something for around £30 max I would have thought. You can always keep the original FLAC files for backup (I do), the converted files will sound the same as the original if you use a lossless format (Apple Lossless being the obvious option). If you have a lot of FLAC files make sure you get software that can do batch conversions - bit of a pain but at the end of it you have something that is more easily managed within one player.
Good luck