Here's what I would do. Choose any point on the circumference of the circle (drum). Hold your ruler fixed there and "swing" it slightly on the other side, pivoting on your fixed point. The opposite point on the circumference will be at the shortest distance from the fixed point. Mark that point and run a string between the two points, marking a diameter. (Another way to do this is to use a tape measure to measure the length of the entire circumference, then measure half that distance from the first point. But I don't think that is quite as accurate.)
Now choose a new point and do the same thing. For the most accuracy this new point should be well away from the previous two points. Half way between them, around the circuference would be good but being accurate with that is not necessary.
The point where the two strings (diameters) intersect is the center of the circle.