Learn German: Ich muss morgen früh aufstehen. - I have to get up early tomorrow.

Modal verbs like müssen (must/have to) are irregular. For ich (I) and er/sie/es (he/she/it), they do not take the usual personal endings. So it is ich muss, not ich müsse. They also often change their vowel, like the ü in müssen becoming a u here.

Why is aufstehen at the very end of the sentence?

When you use a modal verb like muss, it takes the normal verb spot (second position). The main action verb (aufstehen) gets pushed to the very end of the sentence in its original dictionary form (infinitive).

aufstehen is a separable verb. Why isn't it split up here?

You are right that it is separable! Normally, you would split it: Ich stehe auf (I get up). But because the modal verb muss pushes the action verb to the end of the sentence, it stays whole. Separable verbs only split when they are the main conjugated verb in the second position.

How do I know morgen means 'tomorrow' and not 'morning' here?

The lowercase morgen means 'tomorrow'. If it meant 'morning', it would be capitalized as a noun: der Morgen. The phrase morgen früh is a set expression that means 'early tomorrow' or 'tomorrow morning'.