Competitive Super Smash Bros. Melee player here that's getting into speedrunning.
I highly recommend getting a CRT over an LCD, especially if you're looking to get serious in speedrunning a game with frame-perfect inputs. The slightest bit of display lag from an LCD display can be enough to throw off timings and make some tricks difficult.
Display lag is what happens when an LCD tries to take in a signal it's not familiar with and tries to display it on it's screen. Think of it like this: LCDs are digital video displays and N64's composite an analog video output. The LCD doesn't natively support composite signals of video, so it has to process it before it displays it. Processing the video feed creates display lag in itself. Some LCDs even try to upscale the resolution and this will cause even grater display lag. Newer consoles like the Wii U natively output their video feed to HDMI, which LCD screens take in without having the process the signal. They are both digital, so there is no problems there. Older consoles that lack HDMI and favor analog feeds like composite (NES/N64/GCN/etc) will only be able to play lagless on a CRT without heavy modifications done to the system itself. tl;dr: You'll want a CRT for Mario 64 speedrunning since it's an N64 game.
If you're worried about finding a CRT, don't worry - CRTs are quite easy to come by. You can pick them for free or for a couple of bucks just about anywhere. Look around thrift stores like Goodwill or in dumps. I once found a Sony Trinitron on the side of the road that I was able to salvage, and it works perfectly.
There's a lot more information on display lag/LCD lag in general on MIOM. I suggest you check it out. Let me know if you have any questions on display lag and whatnot and I'll try to answer them with the best of my ability. http://www.meleeitonme.com/this-tv-lags-a-guide-on-input-and-display-lag/
EDIT: FYI since you're in Sweden and your region has different A/V standards - When looking for a CRT make sure you find one that will display games in 60 hz. As far as I'm aware some of your CRTs might only support 50 hz, so the video feed can be faulty (I've seen the games feed display in black and white when being displayed on a 50 hz CRT TV.) if you try to play on them. Just keep that in mind.