Learn Italian: È oculato risparmiare piuttosto che pagare per ogni piccolo sfizio. - It is prudent to save rather than pay for every little whim.

This is an implicit alternative clause. When you are contrasting two verbs and the subject is the same (here, it's a general, impersonal subject), you use piuttosto che followed by the present infinitive (pagare) to mean "rather than [doing something]". It elegantly balances the two alternative actions: risparmiare and pagare.

I've heard some Italians use piuttosto che to just mean "or". Does it mean "or" here?

No, here it strictly means "rather than" or "instead of". While it's true that in modern colloquial Italian (especially in the North) people often use piuttosto che as a synonym for o (or) to list equal options, standard grammar considers that incorrect. In this sentence, it is used traditionally and correctly to indicate a clear preference for the first alternative over the second.

What is the literal connection behind the word oculato?

Oculato translates to "prudent", "cautious", or "shrewd". It shares a Latin root with occhio (eye) and the English word "ocular". You can think of an oculato person as someone who is observant, careful, and "keeps a close eye" on their money or decisions.

Is sfizio a common word, and how else might I hear it used?