Depending on practice and skill level you might be able to achieve a fairly even surface on the inside from the tool.
If you are using a 3/16" cutter it is going to be rough ( until you get lots of practice ), a round carbide ( like a Hunter tool ) can leave a very nice surface, other scrapers will usually fall somewhere in between.
As far as what to do next,there are a couple of schools of thought.
One is to sand the inside reachable areas and perhaps the bottom. Places where an average person would see through the opening or feel.
Another is to make a collar of some nice wood that has a small opening that a finger would not fit through and leave the inside rough.
Still another would be to apply black gesso on the inside to make it 'dark and mysterious' and leave it mostly rough.
And finally try your best to sand the interior after carefully using a larger scraper to even the high spots out.
A few folks advocate putting some rough gravel or other sharp edged material on the inside, sealing the hole and letting the lathe turn at low speed for a while to "sand" the inside. Not sure how that works out.
Just my $.02.