Learn Polish: Tatuator raczej nie ma dzisiaj czasu. - The tattoo artist rather does not have time today.

Not quite. While raczej can express preference in other contexts, here it acts as an epistemic particle meaning "probably," "most likely," or "I'd say." It shows that the speaker is fairly confident the tattoo artist is busy, but leaves a tiny bit of doubt.

Why is the word for time in the Genitive case (czasu) instead of the Accusative (czas)?

This is the classic rule for negated verbs. The verb mieć (to have) normally takes a direct object in the Accusative case (for example, On ma czas - He has time). But whenever a verb that takes the Accusative is negated with nie, its object automatically flips to the Genitive case (On nie ma czasu).

Could I put raczej at the beginning of the sentence, like Raczej tatuator nie ma dzisiaj czasu?

You could, but it changes the flow slightly. Placing raczej right before the negated verb (raczej nie ma) is the most natural spot because it directly modifies the state of "not having time." Putting it at the very beginning emphasizes your assumption right away.

I know dzisiaj means "today." Could I use the shorter form dziś in this sentence?

Yes, absolutely! Dziś and dzisiaj mean exactly the same thing and are completely interchangeable. Dziś is just shorter and very common in casual, spoken Polish.