A ballpark check is pretty simple to point you to gauge or sender. You do need a meter to test the sender, but only a wire to test the gauge. Which you do first is up to you. If I don't have a suspect, I go easiest first. You will only will need an additional wire long enough to go from a good ground or battery negative to the top sender connection to perform the temperature gauge diagnostics. That nice shiny bolt in the bottom left of the picture is likely a good ground (it doesn't look like a good connection where the dvm lead appears to be on the solid paint in the picture). Not to repeat myself, but the ground has to be good for the testing to be valid. A small gauge wire is all that is needed, like alligator clips, or "normal" gauge(18-26) speaker wire lying around. Bigger is OK, just sometimes harder to work with. Whatever wire you have around.
Since the sender is bolted into the engine, it's base is in a good ground. The principle is that a good sender will decrease resistance(ohms) as it heats. Cold, the resistance is high, but will decrease as the sender heats. This decrease in ohms is detected by the temperature gauge. So when the sender wire connected to the gauge has no resistance(and is at battery/engine ground), the temperature gauge should read full scale hot.
Testing temp gauge: engine not running for safety, but gauges powered on. Connect aforementioned well grounded wire to cleaned top sender connection. If the temp gauge reads hot=gauge, wire to gauge, and connections at gauge are good. If temp gauge doesn't read hot, one of the three is bad.
Testing sender: engine off and cold. disconnect existing sender wire from top of sender. Connect one lead of dvm to top of sender, the other lead to a good ground. Since your meter doesn't auto-range, move the dvm selector knob to the smallest ohm setting that doesn't give a over range display(20k like you had is likely ok). record the reading on the dvm. Verify there is no risk of wires or anything getting tangled, start and run the engine till warm. as the engine is running, watch the digital ohm meter. If you do not see anything that is more than aprox 200-400 ohms in difference, the coolant temperature sensor is defective and will need to be replaced. I am remembering the ohm values from a long time ago, but if the temp gauge tests ok, it is likely the sender. I can take measurements on mine tommorrow, which is a 5416, aka m20. I would be surprised if the sender is different.