Yes, that is most likely the reason for the blood. The lining of the nose is very thin, and there are blood vessels just beneath the surface, so if you irritate or scratch the inside of your nose it will bleed. He'll be fine, but you should probably try to use saline in his nose first to loosen the mucous, that way you might not need to use the bulb so much. As long as he can breath and there's not a ton of snot running everywhere, I'd leave it alone for a little while so it can heal. Hope your baby feels better soon!
edit: oops, you said you were using saline, sorry!!