Learn Romanian: Fac cumpărături online mai puțin des decât el.

Yes, absolutely. Mai rar literally means "more rarely" and is the most common, natural way to say "less often" in Romanian. Mai puțin des is also grammatically correct and means exactly the same thing.

Why do we use decât here for "than"? Can I use ca?

In Romanian comparisons involving "more" or "less", both decât and ca are correct. Decât is the standard, more formal choice. Ca is very common in everyday speech, so mai puțin des ca el would also be perfectly natural here.

English uses the object pronoun "him" (than him). Why does Romanian use the subject pronoun el (he)?

In Romanian comparisons, you use the nominative (subject) form of the pronoun when comparing two subjects. The sentence is essentially a shortened version of "than he does it" (decât o face el), so you must use el rather than an object form.

Why use the phrase fac cumpărături instead of just using the verb cumpăr?

A face cumpărături is a set phrase meaning "to shop" or "to do the shopping" as a general activity. If you used the verb a cumpăra (to buy), you would typically need to specify exactly what objects you are buying.