State-elimination on FSA, after epsilon-removal construction
Should I first follow the epsilon-removal construction, and then do state elimination?
  Or can I use state-elimination directly on this FSA?


If you would like to do more work to kill time, or if you would like to check the effect of epsilon-removal construction, or if someone higher on the command chain forces you to (for reasons I have no informed guess about), you should do the epsilon-removal construction before state elimination.

Otherwise, it is perfectly fine to start state-elimination directly and immediately.

By the way, the regular express I got is 
$$\left(0+(0+02)(2+42)^*(4+\epsilon)\right)^*(0+02)(42+2)^*\,.$$