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

Yes

The word "curso" is identical in meaning, spelling, and grammatical function in both Brazilian and Continental Portuguese. The only difference between the two variants is pronunciation. In Brazilian Portuguese, the vowels are generally more open and the rhythm is "syllable-timed" (each syllable has a similar duration). In Continental Portuguese, vowels—especially unstressed ones—are much more closed or even reduced/omitted, resulting in a "stress-timed" rhythm.

Brazilian Portuguese

1. Eu estou fazendo um curso de administração. (I am taking an administration course.)
2. O curso de inglês é muito legal. (The English course is very cool.)
3. Ele se matriculou num curso de desenho. (He enrolled in a drawing course.)
4. Preciso terminar meu curso logo. (I need to finish my course soon.)
5. Fiz um curso de informática no mês passado. (I took a computer course last month.)

Continental Portuguese

1. Estou a fazer um curso de gestão. (I am taking a management course.)
2. O curso de inglês é muito fixe. (The English course is very cool.)
3. Ele inscreveu-se num curso de desenho. (He enrolled in a drawing course.)
4. Tenho de terminar o meu curso depressa. (I have to finish my course quickly.)
5. Fiz um curso de informática no mês passado. (I took a computer course last month.)