The carb on this GCV160 engine is very inexpensive (<$20) and I'd suggest cleaning out the fuel tank/tubes really, really good. Give it a good bath, install a new carb, gaskets, change the oil (10W-30) and probably a new spark plug. I'd guess the blade(s) is shot as well, so a new or re-sharpened one would be ideal.
With the serial number off the side of the engine (GXXXX-1234567) any Honda dealer can get you engine parts. You'd need to work with MTD (the actual mower mfg.) to get the blade and non-engine bits.
Here's where to find a Honda dealer who can get you parts:
Find A Honda Dealer