Slip rings are a straightforward solution to the problem, so long as the electrical noise they generate is small compared with the signals you want to measure.
Since you are already using an Arduino, you might consider an alternative, which is to have a second, battery powered, Arduino attached to the rotating drum itself, and use that to transmit the data to an external computer wirelessly, for example using a Bluetooth data link.
I'm assuming the composting bin only rotates slowly, to mix up the contents, so there would not be any mechanical problems with a relatively large unbalanced mass (i.e. the Arduino) attached to it. But this "rotating transmitter" concept is used successfully in situations where the environmental conditions are very hostile (e.g. rotational speeds of the order of 10,000 RPM, high temperatures, and a corrosive atmosphere which would quickly destroy slip rings) - using purpose-built transmitters though, not an off-the-shelf Arduino.