Learn Polish: Jutro muszę wcześnie wstać. - Tomorrow I have to get up early.

It is built right into the verb! Muszę is the 'I' form of the modal verb musieć (to have to / must). In Polish, the ending of the verb tells you who is doing the action, so dropping the pronoun ja (I) is completely normal and sounds more natural.

Why is wstać left in its basic dictionary form?

This is a key rule for modal verbs like musieć (to have to), móc (to be able to), and chcieć (to want). You only change the first verb to match the person (muszę = I have to), and the action verb that follows stays in the infinitive, or dictionary form (wstać = to get up).

Can I put wcześnie at the end of the sentence, like we do in English?

Yes, you can! Polish word order is quite flexible, so saying Jutro muszę wstać wcześnie is perfectly correct. However, putting the adverb (wcześnie) right before the verb it describes (wstać) is a very natural and common pattern in Polish.

Does wstać mean the same thing as 'to wake up'?

Not exactly. Wstać specifically means 'to get up' (out of bed) or 'to stand up' (from a chair). If you open your eyes but stay in bed, you have woken up, but you have not actually gotten up (wstać) yet!