Without some more detail (serial number, model if you know it, number cylinders) it's hard to help verify the part. I think you have a Mercury "800" but they made 80HP in 3, 4, and 6 cylinders over the years so hard to know which. I'm pretty sure though if you just go to the Mercury parts finder for your engine (on this site) you will be able to either find the fuel pump aftermarket or the part number at least. Armed with the part number, if all else fails and it's no longer produced, you can go on Ebay and like 99% of the time someone has either the exact brand name part in old stock or an aftermarket one for that application.
As far as whether you need the part - what you describe is classic fuel pump trouble-shooting. By manually pumping the bulb you are acting as the fuel pump. When you rebuilt it I take it you did not replace any parts? Most likely you've got a bad valve, diaphragm, or a leak that keeps it from working right. Start there.
You can verify the pump is working by disconnecting the output and cranking the engine, it will give a nice pulse of fuel output.
Jon