Acting up

Yesterday my car started to act up. It’s very unusual and inconsistent. I had a hard time starting the car (the starter would spin but the car would not start). Then on my way to work the RPM meter dropped all the way to zero and up.

I am starting to think that may be the fuel pump is going out. What do you think?

All comments and advice would be appreciated. Also, is their a way I can check the fuel pump without buying expensive equipment?

Thanks in advance,