Q. If you’re following Chicago’s rules for title case (at CMOS 8.160), do you capitalize short prepositions that aren’t immediately followed by their objects? A few examples that have come up in my work recently (anonymized a bit and left lowercase for you to resolve) are “paid for by grandma,” “what will grandma dress up like?,” and “whom has grandma turned to for advice?”
A. In your examples, up looks more like an adverb—one that modifies the verb dress—than like a preposition. So let’s call it an adverb.*
But for, by, and to are more clearly prepositions (as is like, though as the last word in a title it would always be capped). So:
Paid for by Grandma
What Will Grandma Dress Up Like?
Whom Has Grandma Turned to for Advice?
It doesn’t matter that the object of for in the first example is unstated or that the object of to in the third example comes first. A preposition is a preposition (except when it’s not).
__________
* The entry in the OED for “up (adv. 1),” sense II.18.c, cross-references several verbs that form phrasal verbs with up, including dress.