So my car would barely start today...

I have been so busy lately with rust repair on some 1972 Toyota Land Cruiser doors, working at my daughter's house and catching up on powder coating orders (good thing I am retired), that I have not started my car in about a month. Lately, when it hasn't been started for a while, it has taken much longer than usual to draw gas into the carburetor. It used to only take 20 seconds or so. Today, I eventually had to squirt some starting fluid into the carb, because the battery was starting to go flat. It finally started and ran fine. I was wondering if a mechanical fuel pump can get weak enough to have trouble drawing fuel from the fuel pump, but be strong enough to keep fuel flowing once the engine starts? putting in a new fuel pump is no big deal, I just don't want to do it if it won't help.