Arduino as switch for external circuit
Three-position makes it a little more tricky than normal.

The simplest option, though depending on what the circuit is, maybe not the safest option, is to use three relays.



Switch only one relay on and it is the same as having the three-position switch in that same state. Switch that relay off and switch another on and you have changed the switch position.

The danger is that this arrangement can have two or even three relays on at once, something that can't happen with a switch.  Also how you do the switching could be critical. Switch one relay off, then add a delay before switching the next one on, so that the previous one has a chance to switch off properly first. That way you avoid having two on at once. Of course, depending on what it is you're switching, that may or may not be an issue.

The safest option, though the hardest to create, would be to physically move the existing switch with, say, a servo motor. Without knowing a little more about the mechanics of your switch I can't really advise more on this option.