St. Petersburg is wonderful...but can be pricey, so finding cheap accommodations could prove difficult. What does "cheap" mean to you, money-wise?
Here's a hotel that's right on the water with decent prices. Different hotels have different "high season" and "low season" periods that April is usually right on the fringe of, so you *might* be able to save a little bit of money. If it's more than just you going (like say...4 people or so), you might want to look into apartments, which can be a better deal for more people. If it's just you then that might not be necessary.
And since you say you're planning on visiting some famous theatres and museums, I assume you're already planning on seeing the Hermitage, so I won't feel obligated to push that idea at you ;)