Learn Russian: Это было типа шутки.

Here, типа acts like a preposition meaning 'a type of' or 'a kind of'. Just like the noun тип (type) would take a genitive noun after it, the colloquial word типа forces the following noun, шутка (joke), into the genitive case (шутки).

We are learning about мол and якобы for hearsay. Does типа work the same way?

Not exactly. While мол and якобы show that you are quoting someone else or doubting a claim, типа is used to show approximation, meaning 'like' or 'sort of'. However, in very slangy speech, young people sometimes use типа to introduce a loose quote or thought, similar to how English speakers use 'like' (for example: And he was like...).

Why is the verb было in the neuter form, even though the word for joke (шутка) is feminine?

Because the grammatical subject of the sentence is the neuter pronoun это (this/it), and the core structure is это было (it was). If you used типа strictly as a slang filler word instead of a preposition, you could say Это была типа шутка, where the verb была agrees with the feminine noun шутка.

Can типа be used as a filler word without changing the case of the following noun?