Nokia 1208 Sim Card | Phone/Device Issues and Features | Consumer forum

Yes, it's possible. The picture though is not a Nokia 1208. It looks like a Nokia 1661.

All you have to do is push down on the little arrow, as you said, and slide the sim out. It take a little work, but it does slide out. That picture is the same set-up as my Nokia 5130. It is awkward.
As for where the sim card goes in a Nokia 1208, it is just a little latch. You place the sim on the contacts, then press the latch down, locking the sim in place.