Learn Russian: Официант, можете налить мне рюмку коньяку? - Waiter, can you pour me a shot glass of cognac?

This is the Partitive Genitive (also called the Second Genitive). For certain uncountable masculine mass nouns like коньяк (cognac), чай (tea), or сахар (sugar), we can use the ending -у or -ю instead of the regular -а or -я when talking about a quantity or a part of the whole. Here, you are asking for a specific amount: a shot glass (рюмку) of it.

Could I still say рюмку коньяка instead of рюмку коньяку?

Yes, you absolutely can. The regular genitive ending -а (коньяка) is always grammatically correct. However, the partitive -у ending is very common in spoken Russian and sounds especially natural when asking for portions of food or drinks.

Why is рюмка (shot glass) changed to рюмку here?

It is in the accusative case because it is the direct object of the verb налить (to pour). You are asking the waiter to pour 'what?' - a shot glass.

Why use the perfective verb налить instead of the imperfective наливать?

You are asking for a single, specific, completed action: you want one shot glass of cognac poured and given to you right now. For a one-time request focused on a completed result, Russian uses the perfective aspect.