IF the chain is actually worn past 1%, and that was the sprocket you used the most, then it's not surprising that it's the most worn ans skips with the new chain.

In your shoes, I wouldn't buy a new cassette yet. The skipping is caused by the chain bumping the tooth's corner as it swings in. If the skipping isn't too annoying, and you can ride this sprocket with reduced power for a while, the chain will wear that corner fairly quickly and solve your problem by itself.

Otherwise, you can remove the cassette, and disassemble it to free that sprocket, then gingerly grind a tiny bit off the back top corner of each tooth. You want to only remove a tiny bit (worth repeating a TINY bit of material) from each corner. If you own a bench grinder or Dremel tool, you can use either. Otherwise you can do the job with a coarse emery board. (don't use a file, the teeth are hard and will dull your file).

BTW- that kind of chain wear that fast is a bad indicator. Unless you do tons of rain riding, steep hill climbing, weigh a lot or are a very powerful rider, you should do much, much better. I suggest you review your choice of chain lube, and chain care practices (consider the source, on the chain lube/chain care advice).