It's winnable, but you need to be careful and have practice. If you get blown up, you lose. Not exploding is the key to winning this.
You might also need to get lucky with the gift boxes. Collect the big nitro at all costs. This might need practice, but it will help you a lot. If you have bubble gum, you can use it to protect you from the first banana. Only grab the small nitros if you can. I think they are too far in the outside and it costs valuable time to try to collect them. Avoid other karts as they are out to destroy you (it's Expert mode, after all!). Head for 1st place as soon as possible. The idea is to get out of the way of the other karts so they can't attack you directly with cakes or bowling balls. Also, it's a lot easier if you manage to become 1st early in the race than later on. The first seconds are often crucial in this game. This is very important. Collect gift boxes. Pray you get a turbo boost. Use these only in the long straight parts, then you have a big advantage. If you get a bubble gum, keep it and use it whenever you hear a basketball approaching. Look back from time to time, just in case. Also learn how to drift. Stay away from the blue/white strips at all costs, they slow you down dramatically and ruin your race.
And then the general rules for time trial races apply, of course. E.g. Drive close to the inner part of curves, not in the outer part. You want to minimize the amount of road you need to drive in order to complete a lap. Also, choosing a different kart might do wonders. And: Practice, practice, practice!
General tip: Only tap the nitro key, don't hold it pressed. You will have maximum speed for a few seconds anyway. Then wait until your speed lowers again. Then you can use nitro again. You will save a lot of nitro this way and still be faster.
BONUS TIP! A well-aimed plunger can be used to destroy a basketball, but it's tricky to do so.
PS: My record in this challenge is 02:34:45 with Emule. I still think this challenge is quite hard.