Q: Is it possible to upgrade 13" MBP from 32bit to 64bit

No. If it has a Core 2 Duo processor then it is 64-bit. If it does not have a 64-bit EFI boot ROM, then it cannot run the 64-bit kernel and would not be able to run Mountain Lion.

Read here:

http://support.apple.com/kb/HT3770

And here:

http://support.apple.com/kb/HT4287

If you purchased a new Mac two months ago it has a 64-bit processor. There's nothing to upgrade. If you want to know if you can upgrade the existing CPU to something else, then the answer is, No. The CPUs are not upgradeable.

http://support.apple.com/kb/HT3696