MY BROTHER IS LEAVING THE CITY FOR A YEAR

Writing the title in all capital letters like this may give the impression that you're shouting loudly. I understand that some texts such as scientific papers may also do this, but I would not recommend it here.

We really get on well (with each other) and we always have fun together.

(1) If you want, you can omit "with each other", because the reader can infer that you're referring to your brother and yourself.
(2) I'd suggest omitting the second "we" to avoid repeating the word.

He's older than me and he will endsoon finish his degree, but he is going to finishdo it in Pragaue.

(1) I don't think "end his degree" is a natural expression.
(2) In English, the city is known as "Prague".

He will do an erasmus the nextjoin the Erasmus programme the following year, it's similar to a scholarship.

(1) "Erasmus" is the official name of a programme and should be capitalised.
(2) I'd suggest stating explicitly that Erasmus is a programme, for the sake of readers who might not know.
(3) I think "the following year" sounds more natural. Alternatively: "He will join the Erasmus programme next year".

Our goverment will paygive him an amount of money and he will be able to gcomplete his career deducatione in another country.

(1) I wouldn't use the word "pay" here, as the government doesn't owe your brother any money in exchange for a service.
(2) "Career" would moreso refer to a job.

I'm really upset because he is like the union of allglue of my family, I feel much more comfortable with my family if he is next to me.

One of the last things we are going to do before he leaves is get a tattoo about us, wof us. We are thinking aboutof some ideas but we are not sure enough for now.

(1) "A tattoo" is just a noun, and is not something you "do". "Get a tattoo" would be the action.
(2) I think breaking this sentence up into two sounds more natural.
(3) The phrase to use is "think of (something)". "Think about (something)" also exists, but isn't used here.