Learn Icelandic: Ert þú búinn með rúlluna? - Are you finished with the roller?

In the set phrase vera búinn með (to be finished with a task or consumable), the preposition með always takes the accusative. If you used the dative here, it would completely change the meaning to 'Are you equipped with the roller?' rather than 'Are you finished with it?'

How do we know the gender of the person being spoken to in this sentence?

The word búinn is the masculine singular form. Because it acts as an adjective describing the subject þú (you), we know the speaker is talking to a male. If they were talking to a female, they would say Ert þú búin með rúlluna?

Does rúlluna strictly mean a paint roller here?

It depends entirely on context. Rúlla can be a paint roller, a roll of tape, or even a toilet paper roll. The phrase vera búinn með works perfectly for both finishing the use of a tool (like painting) or using up a consumable supply (like paper).

Could I use the verb að klára (to finish) instead of this phrase?

Yes, you could say Hefurðu klárað rúlluna? (Have you finished the roller?), but vera búinn með is the most natural, idiomatic way in spoken Icelandic to express that you are done using a specific tool or have consumed an object.