Is the word neste the same in Brazilian and European Portuguese?

Yes.

The word neste is a contraction of the preposition em (in/on/at) and the masculine singular demonstrative pronoun este (this). There are no differences in meaning, spelling, or grammar between Brazilian and Continental Portuguese. The only difference is pronunciation: in Brazilian Portuguese, the unstressed vowels are often reduced or neutralized (the "e" often sounds like a short "i"), whereas in Continental Portuguese, the vowels are more clearly articulated or follow different phonetic patterns.

Brazilian Portuguese Examples:

1. Neste livro, encontrei a resposta. (In this book, I found the answer.)
2. Neste final de semana, vou descansar. (This weekend, I am going to rest.)
3. O que tem neste pacote? (What is in this package?)
4. Neste carro, não cabe todo mundo. (In this car, everyone doesn't fit.)
5. Neste lugar, é muito legal. (In this place, it is very cool.)

Continental Portuguese Examples:

1. Neste livro, encontamente a resposta. (In this book, I found the answer.)
2. Neste fim de semana, vou descansar. (This weekend, I am going to rest.)
3. O que é que há neste pacote? (What is in this package?)
4. Neste carro, não cabe toda a gente. (In this car, everyone doesn't fit.)
5. Neste sítio, é muito giro. (In this place, it is very nice.)