I usually tip the enclosure lightly and chase them out carefully into the new one, I clean mine once every 6 months, but note, I spot clean and use a mix of plantation soil and vermiculite.
Sure you can put cork in there, just a heads up, spiders are ultimate destroyers. So far every tarantula I owned chewed pieces out of the cork, it's really funny to see.
If he's that small you can even use your hand to move him, like I do with my little sling right now, just be warned, they can be FAST.
I'l take a picture of my enclosure and post it in a minute or two, I'l also add a picture of my sling, since he's currently in hiding (suspect he has to shed).

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And yeah, my windows get dirty instantly when misting, he's been in there for around a month or two.