Removing glued on material from this subwoofer
I got this picture from the same source as the one included in the question.  It shows more of the internal detail and gives a better look at the covering.  I left it at size large so the detail could more easily be seen.

The tube is definitely made of wood, my guess would be either bent plies or straight barrel staves.  In any event it looks durable.

The covering looks like it is a woven fabric that is skillfully cut and glued to the barrel.  See how it is folded into the opening and reveals some unevenness at the edges where it has been cut to fit the smaller diameter of the inside as compared to the outside of the tube.

My recommendation for removal of the fabric is to use a heat gun that will soften the glue and allow you to peel off the fabric. Easier said than done because the fabric will quite likely tear and require the use of some sort of scraper. 

If the glue becomes solid after cooling from the heat treatment you can sand it off.  If it remains tacky, you can use some sort of solvent as @matt suggests.  You definitely want to remove the glue because if not, you will have a rough, irregular surface that will be telegraphed through your shiny black paint - even if it's a thick layer.