If it is installed similar to a vintage Fender, you should be able to remover the speakers, the unscrew the baffle (from inside the cab), and it'll pop off the front. I'd not take the cloth off the baffle, but clean as best as you can, dry and put it back.
air/hair dryer? If sticky, I'd probably use a wet paper towel to sop up the stickiness. Never had this problem, but logic leads me to try this. Don't soak the cone, though. I'd probably just deal with a stain. But a moist towel might get some of the stickiness off. I just repaired a tear in the paper surround of a vintage speaker, and air dried the repair within about 8 hours or so. I wasn't in any hurry, so I just let it dry on its own.