Measuring an unknown world / solar system

[Excession]

For the mass of the planet, you'll need a stopwatch and a ruler (and the planet's approximate diameter, which Farothel already told you how to get). You'll want to measure the acceleration a dropped object experiences at 'ground' level, and measure again at a higher (or lower) elevation. Using the difference between those two accelerations you can calculate the mass of the planet.

A pendulum should be a more accurate way of measuring gravity. You'll still need a ruler and a timing device, but you won't need to be as precise on the timing. To avoid modern technology use a pocket watch with a torsion pendulum, or your own resting heart rate. Changes in fundamental constants will screw this up though. If mass changes then the torsion pendulum will be off, and any such change might be quite exciting (or fatal) so will mess with your resting heart rate.