600 USD is almost what I spent on my camera, so it won´t make sense to spend that amount on a mic..
Just for reference, on a big-budget set you may find a camera kit worth $50k, but the sound mixer likely has $250k tied up in the sound kit. There is a flawed thinking that the sound gear should be cheaper than the camera. You've said this is a hobby-level thing, which I certainly understand, but I just wanted to give you a little frame of reference here.
For $200 there's not much you can do that's worthwhile because it takes more than just the mic. Start with the mic, add a boom pole and suitable shockmount, plus some form of wind protection (basic foam may be fine indoors, but outdoors will need much more). And, you'll HAVE to have headphones. Never record sound without monitoring, else you should prepare yourself for disappointing surprises in post.
HERE is a decent shotgun mic with included shock mount, which is absolutely about $15 above your range. This does not include cables to connect to your camera, and because of the nature of your camera you'll need to take the mic through a mixer or pre-amp (more $). Plus, there's no boom pole included.
I would not recommend recording to a computer, though if you did that it would still require the proper interface (USB-based, with built-in pre-amp)... at least $200 for that alone.
There is
THIS, though, that may do you fine for your hobby. You'll need to get an extension cable and a boom pole, but be prepared for hobby-level sound results. Anything is better than on the camera, but this one has its limitations. Because of the 1/8" cable running an unbalanced signal, the more you extend the cable run the more you risk degrading the signal. But again, it may suit your needs.
For $45/day, you can rent a wireless kit here in the States. I don't know what the going rate (or availability) would be in Norway. With the proper 1/8" output cable avilable for systems like the Sennheiser G2, you can plug straight into your camera's mic input.