Bicycle's inner-tube got puncture right next to the valve, How to patch it there?
That's a very hard area to patch properly (if its even possible), and I'd recommend putting a new tube in instead of trying to patch it. . 

I'd also check that the rim tape on the rim is intact and in good condition, cause otherwise if its busted, you're going to get another cut. Also, as pointed out by ChrisH in the comments, rough edges on the rim hole can also cause this problem (you can use a small file to smooth them out, if that is the case).