Learn Russian: Ладно, я работаю.

Russian does not have a separate continuous tense like English does. The single verb работаю translates to both "I work" as a general habit and "I am working" right at this moment. You rely on context to know which one is meant.

Is ладно used like an annoyed "fine!" in English?

It can be, but usually it is just a very common, neutral way to say "okay," "alright," or "agreed." Here, it acts as a conversational transition, similar to saying "Alright then, I am working."

Why does the verb end in -ю?

This is the regular present tense ending for я (I). To make the "I" form for most verbs ending in -ать, like работать (to work), you remove the -ть and add -ю.