Learn Romanian: Eu citesc acel document acum. - I am reading that document now.

The word acel is used for masculine and neuter nouns in the singular. The word document is a neuter noun, so it acts like a masculine noun when there is only one. If the noun were feminine, like carte (book), you would use acea instead.

Can I put the word acel after the noun?

When you use the short forms like acel or acest, they must come directly before the noun. There is a way to put "that" after the noun in Romanian, but it requires adding "the" to the end of the noun and changing the demonstrative word slightly (to make documentul acela). Putting it before the noun is perfectly natural and easier to remember at this stage.

The English translation says "am reading", but isn't citesc just "I read"?

Romanian does not have a separate continuous tense like English does. The single present tense verb citesc translates to both "I read" and "I am reading". In this sentence, the word acum (now) makes it extra clear that the action is happening right at this moment.

Do I have to include Eu at the beginning of the sentence?

No, it is optional. The verb citesc is specifically the "I" form of the verb a citi (to read). Since the verb itself tells us who is doing the action, you can safely just say Citesc acel document acum. Adding Eu just gives a little extra emphasis to the fact that you are the one doing it.