Two key things to a successful carb overhaul:
• It must be totally cleaned, especially the main jet and nozzle area. *Any* tiny bit of slime, debris, etc. can cause a problem and the engine won't start.
• It must go back together exactly right, and all gaskets must seal up tight. One screw not set right, one part not secure, or a tiny gap or tear in a gasket will also prevent the engine from not starting.
Sometimes, it's faster-cheaper-easier to simply replace the carb and install new gaskets. Depending on the engine, a new carb and gasket set can run less than $30, and if you spend 1-3 hours trying to get an old carb working again, it may be wiser to just replace it.
If you will provide the serial number off the back of the mower deck, I can tell you the part numbers and list prices for a new carb and gasket set. Include your ZIP code and I can list out dealers close to you who may have the parts in stock.
As others have said, be sure to verify the choke is working properly. It must fully close when the engine is cold, and many times even when hot, for the engine to start easily.
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Caveat: I work for Honda, but the preceding was my opinion alone.
