Learn Russian: Я впервые в этом городе. - I am in this city for the first time.

Both mean 'for the first time' and you will hear both frequently. Впервые is a single adverb that sounds slightly more polished, while в первый раз is a bit more conversational. You can safely swap them in this sentence without changing the meaning.

Why do we use этом instead of этим here?

Because we are talking about a location (answering 'where?'), the phrase в этом городе must be in the prepositional case. Этом is the prepositional form of the masculine word этот (this). Learners often mix it up with этим, which is the instrumental case form.

Could I change the word order and put впервые at the end?

Yes! Russian word order is very flexible. Saying Я в этом городе впервые is perfectly natural. Placing впервые right after Я just puts a slight emphasis on the novelty of your visit.

How would I change this sentence to say 'I was in this city for the first time'?

Because the present tense verb 'am' is omitted in Russian, you just need to insert the past tense form of 'to be' to change the tense. You would say Я впервые был в этом городе (if you are male) or Я впервые была в этом городе (if you are female).