Although I have never been to Universal, I would suspect that the times will be available on the map or message boards around the park.
You must be a very good mommie.
This is what Frommers had to say about it:
Attraction
A Day in the Park with Barney
Set in a parklike theater-in-the-round, this 25-minute musical stars the Purple One, Baby Bop, and BJ. It uses song, dance, and interactive play to deliver an environmental message. This could be the highlight of the day for preschoolers (parents can console themselves with their kids' happiness). The playground adjacent to the theater has chimes to ring, tree houses to explore, and lots to intrigue wee ones. The theater is air-conditioned, but even in the hottest months that fact would not be enough to entice us to spend another instant around Barney.
Source: Frommer's Walt Disney World & Orlando 2003