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

No

While the dictionary definition and spelling of "custa" (the third-person singular of the verb custar) are the same in both dialects, there is a difference in typical usage. In Brazil, the verb custar is frequently used as a primary way to express that something is difficult, hard, or requires a great deal of effort (e.g., "custou muito para ele"). In Portugal, while custar is used for price, when expressing difficulty, speakers much more naturally prefer to use the verb ser (to be) with adjectives like difícil (difficult) or the expression ter dificuldade em (to have difficulty in).

Brazilian Portuguese Usage

1. Quanto custa esse celular? (How much does this cell phone cost?)
2. Custou muito para ele terminar o trabalho. (It was very hard for him to finish the work.)
3. Custa muito para mim acordar cedo. (It is very hard for me to wake up early.)
4. Esse relógio custa uma fortuna. (That watch costs a fortune.)
5. A criança custa a dormir à noite. (The child struggles to sleep at night.)

Continental Portuguese Usage

1. Quanto custa este telemóvel? (How much does this mobile phone cost?)
2. Foi muito difícil para ele terminar o trabalho. (It was very difficult for him to finish the work.)
3. É muito difícil para mim acordar cedo. (It is very hard for me to wake up early.)
4. Este relógio é caríssimo. (This watch is very expensive.)
5. A criança tem dificuldade em dormir à noite. (The child has difficulty sleeping at night.)