You obviously need to be quite skilled for this, and prior experience with the original version will definitely help. I consider myself pretty good at the game yet not mind blowing, so I'm gonna try and give a tutorial for each level based on the minimum skill needed to beat Oxide's time on it. I'm mainly going to focus on the CTR levels as the CNK tracks are far easier barring one which is probably close to the hardest time. All of the new tracks post release don't count toward it and neither does Retro Stadium so I'm not going to cover them.
Anyway it's a given at this point that you can turbo boost while power sliding so I'm working under the assumption that you're reasonably experienced with that mechanic. If you're not then you don't really have a hope of getting this just yet and need to improve your skills. I'm also assuming that you're using a Speed engine for this as once you master it you have a much wider margin for error when going for the times. The lower turning stat also allows you to take turns much tighter and maintain a straight line when drift boosting, not to mention Oxide's ghost are Acceleration class so racing with a Speed class gives you a distinct advantage when mastered.
Crash Cove: One of the harder times. Oxide has a significant speed boost going into the last lap from the boost pads so you will need to maintain large boost fire (colloquially known as Sacred Fire) at this point as well by drift boosting as you go over the pad then maintaining it via perfect drift boosts for as long as you can. If you can maintain this at the earliest opportunity (end of the first lap) then you're golden. Shortcut from the water is obviously a massive help as well.
Roo's Tubes: Not much to say here. Try and maintain boost pad fire and take the shortcut at the end and you shouldn't struggle too much.
Sewer Speedway: MUCH easier now than when I did it given the shortcut fixes.
Mystery Caves: Taking the shortcut near the start just past the first turtles on your left with large boost fire (and hopping to maintain your speed) helps but isn't strictly necessary. Just go fast.
Coco Park: Surprisingly easy considering it's a straight circuit. Cutting across the turf at the first big turn helps but isn't wholly required. Just go fast and try and preserve large boost fire like everywhere else.
Tiger Temple:One of the easiest ones. I got hit by fire numerous times and still comfortably beat him. Take the shortcut obviously, though.
Dingo Canyon: Oxide's performance here highlights just how poorly programmed this track is in this version of the game. The river section is pretty bad here with the ramps requiring far stricter timing than the original version with not much of a benefit for using them. Stick to the land parts and avoid jumping off the start of them and you should be fine. It's surprisingly easy compared to some of them.
Papu's Pyramid: Not a very difficult one provided you're experienced with the track. Taking the obvious shortcut near the middle and the one at the end of the track will clinch it.
Blizzard Bluff:Taking the lake shortcut over the three laps is obviously mandatory and the shortcut at the start of the track helps as well, but thankfully the annoying fence shortcuts at the end can be safely ignored.
Polar Pass: You should be taking the obvious wall shortcut as standard now, but I wouldn't call the lake skip essential over maintaining blue flame (or Ultimate Sacred Fire) as long as possible. Oxide used the lake shortcut but I ignored it given my lack of consistency with it and still beat him. Maybe aim for it once?
Tiny Arena:A very easy one. There's no shortcuts or tricks to master here; he's really quite slow and very beatable if you're at least reasonable at the game.
Dragon Mines: One of if not the hardest time in the game for a number of reasons. The spiral near the end is the main killer obviously, but like Crash Cove he maintains large boost fire from the one pad in the game for the last lap so you will need to do likewise and it helps if you can do it for a couple of laps. The poor programming rears its ugly head again here considering Oxide slams into the wall of the spiral himself on the third lap. Mastering air braking helps immensely for that section. And those bloody mine carts.....
Cortex Castle:Nothing of note here to worry about. Take the end shortcut and maintain blue fire for at least even a little bit into the next lap and you should be fine.
N Gin Labs:I don't think you need to be perfect here but maintain blue fire out of the tunnel for a while obviously helps a lot.
Hot Air Skyway: One of the easier ones surprisingly. Maintaining blue fire off the jump for a little while helps although he is quite slow here.
Oxide Station:Certainly the hardest N Tropy time in the game at least to the point where Oxide's time isn't much better than his, which still makes this one of the harder ones. I used an Acceleration character here for better turning, as you will need to maintain blue fire as soon as you get it to close to the end of the track during at least one of the laps in order to triumph here. Not easy by any stretch.
Slide Colosseum: One of the easiest in the game, if not the easiest of the CTR tracks. Take the jumps and time saves but there is altogether nothing to worry about here.
Turbo Track: Not too hard either although you will need to be at least decent with maintaining blue fire. He struggles with the hairpin turns as well so don't worry too much.
Tiny Temple: This is the only CNK track I'm going to list as it's really the only difficult one, but it's one of the most difficult in the game. You will need to maintain blue fire as soon as you get it for practically the entire rest of the race as that is what Oxide does. You might want to switch to an Acceleration character here just for the better manoeuvrability. Maintaining it is the same principle as with regular large fire (perfect drift boosting).
Anyway I hope this helped you get an idea of the minimum skill level you need to beat these times.