Definitely do take the whole thing apart to clean it. I had the same problem you did after rebuilding my 350F master cylinder with not getting the MC to push fluid, just air. What I ended up doing was "reverse priming" my MC by using a syringe and squirting some brake fluid into the cylinder from the end where the brake line goes on. I then did the "put your finger over the hole" trick someone's already mentioned. Eventually it did start pumping fluid after I reverse primed it. That trick may help you out as well.
Also, there is a Honda service bulletin out (posted somewhere on this site, I downloaded it from here) that states not to pull the lever any more than 15mm away from the handgrip, that over-stroking the lever will force the guts of the MC to a position where it may leak fluid out the lever end of the MC. So, heads up on that...I just take a nut with about 15mm thickness and tape it to the grip while I'm bleeding so I don't oversqueeze the lever.
Keep at it, you'll get it. And as a side note, once you do get everything bled, don't be surprised if you end up with a sorta spongy lever. My 40-year-old brake lines were swelling enough to not get a firm lever even though the system was completely bled. Ended up having to get new ones from a local brake line place that had universal ones that we sorted out to fit. Now the lever is hard like it should be.