I was arrested and convicted in 1976 for Robbery. Will this show up if finger printed for a civil service job ?

Yes, it should.

Felonies show up no matter how old they are. You should fully expect this to show up during a background check.

Yes, the finger prints merely first show who you are and if you were ever fingerprinted under another name. That info is then ran though a criminal background check, such as the NCIC report. It will show any conviction you ever had.