Let's take your last paragraph first as it's easiest to answer. You ask:
would an ant on a spinning drum in empty space know if the drum was spinning clockwise or anticlockwise?
and this is more or less what the Kerr metric describes. The answer is that yes you can determine the direction of rotation by measuring the frame dragging. Whether the rotation is clockwise or anticlockwise is ambiguous because an observer rotated 180º relative to you will measure an opposite direction. If you called the rotation clockwise they would call it anticlockwise and vice versa. Which of you is correct is the sort of empty question that wars get fought over.
Anyhow, aside from that easy bit the rest of your question cannot be answered because you haven't defined what a rotating universe is. Specifying a universe is far harder than I'd guess you appreciate. You have to find a spacetime geometry and matter distribution that are linked by the Einstein equation, and that's hard. I would guess that you are assuming a rotating universe would be the same as a static universe with a rotating coordinate system, but that is most emphatically not the case.
The only rotating universe for which we have an analytic solution to the Einstein equations is the Gödel universe. This an infinite homogenous universe where the curl of the matter velocity field is everywhere constant. It is rotating, but it doesn't have a central axis. Any observer anywhere in the universe would observe the universe to be rotating about them at the same rate.
It's easy to tell the direction of motion in a Gödel universe because it's possible to determine your own rotation by measuring the motions of test particles near you. Adjust your own rotation until the universe around you looks static, then you can determine your own rotation rate using the test particles.