I've read that you can go down to 1/30th sec shutter speed without needing a tripod. What do you think?
Only if you dont zoom in, the more you zoom in, the faster the shutter speed you need.
You are going to struggle without a tripod, but if you can find something to attach the monopod to, so it is held steady, that will help.
Can you tie it to a fence post, garden chair or similar? You want to be able to leave it hands off, so you dont cause any movement.
Once set up, use the self timer to give a 10 second delay, this will allow any vibration from pressing the shutter to settle before it fires.
If you are going to zoom in on the moon, use the optical zoom only (20x), this will give an equivalent focal length of 420mm (ish). Using the digital zoom to go even closer may work, but I find it looses a lot of detail on my panasonic compact.
If you are going zoomed in on the moon, I would suggest using Aperture Priority mode with the exposure compensation set to -2 as a starting point and keep the ISO on 100. If that is still over exposed, then switch to Manual mode for a faster shutter speed.
Hope this helps a bit and good luck.