Learn Arabic (Modern Standard): أعمل طبيبا في المستشفى.

This is the visible indefinite accusative case you are learning about! When an indefinite noun describes a role or state, it takes the ـاً (-an) ending. To write this "an" sound, Arabic adds an extra Alif to the end of most words.

The translation says "as a doctor", but I don't see a word for "as". How does that work?

In Arabic, you don't use a separate word for "as" when stating your profession with the verb "to work". Instead, you just use the accusative case. By making the word for doctor accusative (طبيباً), the sentence literally expresses "I work [being a] doctor".

Where is the word for "I" (أنا) in this sentence?

It is built directly into the verb! The verb أعمل (I work) starts with the letter Alif with Hamza (أ), which is the standard prefix for "I" in the present tense. You could say أنا أعمل, but dropping the "I" pronoun is much more natural and common in Arabic.

Why does the word for hospital, المستشفى, end with a letter that looks like a Yaa (ي) without dots?

That letter is called an Alif Maqsura (ى). Even though it is shaped like a Yaa, it is actually a vowel that makes a long "aa" sound. So, المستشفى is pronounced "al-mustashfaa".