Advice for beating it:
For each cup there are certain must win levels and hard ones
Mushroom Cup
All are about easy but the baby course is the best, because you can usually not only get in first but also screw up the computer slotted to be in 2nd everytime. Keep nailing him with items and he will sometimes finish in 4th and 5th which pratically gives you the cup.
Peach's beach and Desert are easy as long as you don't get hit by obstacles.
Toughest Course: Mario course, although it is not that hard, just hardest of 3 since straight course. Overall easy cup to win.
Flower Cup
The two must wins are the bridge and daisy's cruiser. The computer will fumble around on these courses and give you opportunities to win. Luigi's course can be tough but you should at least get in 2nd.
Toughest Course: Waluigi by far, the rest are very manageable but this one can mess you up. Watch out for traps that will push you into the tar (bananas or fake blocks). Try to hit computers on the turns near the tar and it will mess them up more.
Flower Cup is not too hard, but could mess you up if you run into problems
Star Cup
Star Cup may actually be easier then Flower Cup. The city level and DK Jungle are the best for beating the computers. Computers don't handle the curves well in the city and don't go the short cut. In DK Jungle they stay on the path while you don't have to (just go straight down hill, it is better). DK Jungle is hard for beginners but once you learn to handle curves, it is best for beating computers.
The Ice level is also fairly good to beat computer on as well if you can dodge the shyguys and iceblocks.
Hardest Course: Yoshis Course, probably 2nd hardest in game. Yoshi's course looks easy and straight on at times but the curves can kill you and for some reason the computers are always really good on this course (a lot better on the others). Do not be suprised to find yourself in 3rd or 4th during most of the race. For some reason, if you can stay in the race the computers will sometimes fold at the end and allow you to pass them to finish in first or second.
Actually this cup is fairly easy to win, probably easier then the flower cup if you can take advantage of the shortcuts.
Special Cup
There are two levels that at first glance look really hard but once you play them a few times, you can rock the computers at them. Dino level is the best to beat computers on if you can avoid getting hit by the dinosaur and knocked off on that bridge towards the end. The computer takes the outside in the cave, so if you can avoid the gysers and take the inside, you can have a huge advantage. Wario course also has an advantage if you take the jump pad rather then the hole. L and R buttons are your friend here and vital to winning. This course is daunting at first and expect the computers to jump off to early lead but once you get to that pit and the rest of the way, the computers struggle with that area given you an opportunity.
Bowsers palace is not to hard as long as you can avoid turning to hard and bumping into walls and avoid falling in lava near the end. I suggest slowing down near the end.
Rainbow Road: Toughest level in the game, but Yoshis island is close. The computer NEVER FALLS OFF on the harder modes. This means you must almost be flawless, even at expense of speed. Key thing is to just stay in top 3 and your work from the other 3 courses should get you through. If is very possible to get in first if you use your items right or avoid falling.
Tough cup, but not too tough. Use opportunities wisely and you should beat this one.
