Learn Portuguese (Portugal): O sol nasce no este.

Yes, it does! In Portuguese, instead of saying the sun 'rises', you say the sun 'is born' (o sol nasce). Similarly, when the sun sets, it 'puts itself' away, using the verb pôr-se (o sol põe-se).

I learned that este means 'this'. How can I tell when it means 'east'?

Context is key! As a compass direction, este is a noun. When it means 'this', este is a demonstrative word that usually appears right before a masculine noun (like in este livro for 'this book'). Since it follows no ('in the') here, it's acting as a noun, so it means 'east'.

I've also seen the word leste for east. Are este and leste the same?

Yes, they are completely interchangeable synonyms for 'east'. You will hear both in Portugal, though este is very common when listing the four main compass directions together (norte, sul, este, oeste).

Why do we use no (in the) instead of just em (in) before este?