You should only pick one, instead of using nor and either together. I don't think it's wrong, but it sounds a little strange. "He couldn't get into university" sounds like you're saying he couldn't get into *any* university. This makes the two parts of the sentence have different meanings. You should also use either "could" or "did" in both halves of the sentence, instead of using one in each half. Possible options: "He couldn't get into that university, nor could we." "He couldn't get into that university, and we couldn't either." "He didn't get into that university, nor did we." "He didn't get into that university, and we didn't either." Or, like the above comment suggested "and neither did/could we" is also good! ("...n't either" is considered incorrect, but it sounds natural and is used by native speakers all the time.)

I think you could simplify this sentence a bit (it's unnecessary to have both nor and either here): "He couldn't get into that university and neither/ nor could we."

Hi Behruz. My suggestion: Neither he nor I was able to get into that university. or Neither of us was able to get into that university. or He was not able to get into that university, and neither was I. or He could not get into that university, and neither could I.