First, unplug your mouse before putting anything on it just in case. Grab some rubbing alcohol, paper towels, Qtips, and a can of compressed air if you have it.
Put a bit of rubbing alcohol on a paper towel. Make sure it isn't dripping wet, but a wee bit damp. Give the buttons a good, thorough wipe down. This should cut the stickyness and remove gunk that's on there.
Tilt your mouse sideways so you can see the crack where the button contacts the rest of the mouse. Spray compressed air into there while moving the mouse itself around so you can get all angles. Do this for both buttons, even if only right-clicking is causing problems. Blowing air under the right button may move the dirt beneath the left. If the right-clicking problem was from the stickyness on the button, I'd still clean underneath the buttons regardless.
Use a Qtip on the LED on the bottom of the mouse to remove any dust or hair.