Use number 3,267 for a bandanna: wiping down a wet tarp.
Once I've wiped it down, roll it up and secure it on the outside of my pack (well, due to my pack design, everything is on the "outside", but I assume you have a pack that is more traditional in design). Let the sun dry it if possible, and--if not--wipe it down again when I set up in the evening. Tarps get wet. That's okay; actually, it's good.
When I get home, if the tarp still hasn't dried the whole way, I set it up somewhere out of the weather (my shed is always a perennial favorite) to dry so that it doesn't build up mildew. (Mildew buildup takes weeks, not days, so don't worry about a couple of days with a wet tarp.)
Hope it helps!