I received a string from a web service like this:
CAAAAB+LCAAAAAAABADtvQdgHEmWJSYvbcp7f0r1StfgdKEIgGATJNiQQBDswYjN5pLsHWlHIymrKoHKZVZlXWYWQMztnbz33nvvvffee++997o
This answer here covers Base64.
zlib is available directly as a shared library on Mac OS X and iOS. It can decompress gzip streams.
Note that the first four bytes of your example have to be stripped before you get to the gzip stream. They appear to contain the size of the uncompressed data in little-endian form. (They are 08 00 00 00
in your example.)