Should you do push ups after a bicep workout?
Push-ups work the triceps. It is not ideal to work triceps and biceps on the same day, as it makes creating an effective workout plan difficult. Ideally, you want to be able to rest any muscle group for a couple of days after working it. You can't do that unless you segregate your exercises.
For example, one would typically do bench press (big exercise first, after warm-up), military press, flys, and push-ups on the same day. These are all pushing exercises that work triceps, pecs, deltoids, etc.
The following day, one would maybe do bent-over row, pull-ups, reverse fly, etc. Notice these are all pulling exercises that only minimally require the muscles used the day before. This allows the body time to grow muscle, as triggered by the previous day's workout.
It's important to remember that muscle is damaged (intentionally) during a workout. Growth comes in the days following as the body responds to that damage by repairing and building the muscle back stronger. This only happens if protein is available in the bloodstream to build from, and if the muscle is being rested.
Incidentally, protein doesn't stay in the blood long, so snack on small bits of protein all day, cut carbs to compensate for the increased calories of that snacking, and get plenty of rest... If you can't rest your whole body, at least try to rest the muscles most recently worked out, to the extent that is possible.