Should you hyphenate "well-prepared" when it is used as a verb, as in the sentence "The night of studying has well-prepared me for the test"?
No, because you shouldn't use "well-prepared" as a verb. You might use "well prepared" as an adverb and a verb, analogous to "thoroughly prepared" in "the studying has thoroughly prepared me for the test," but as noted in a comment the idiomatic word order would be "has prepared me well."
To answer the question in the title, you should almost never hyphenate "well prepared." I would do it only to avoid ambiguity, but I can't think of a sentence containing the phrase that would be ambiguous without a hyphen.
Hyphens are useful when a noun phrase is used attributively to modify a noun, as in "green-tea wholesaler." We need the hyphen here to make it clear that we're talking about a wholesaler of green tea rather than a green wholesaler of tea.
In the case of "well prepared," however, we have an adverb modifying a past participle which will then modify a noun: a well prepared student. There's no chance someone would thing that "well" is modifying "prepared student," so there's no need for a hyphen.
As phoog's answer says, "The night of studying has well-prepared me for the test" sounds wrong regardless of whether you use a hyphen or not.
"Well(-)prepared" sounds right in adjectival contexts (where "prepared" is a participle or a participial adjective): "You are well(-)prepared" or "A well(-)prepared student".
There are not very clearly fixed rules about when to use a hyphen between well and a following adjective or participle. In general, hyphenation is preferred more when the adjective or participle is attributive (followed by a noun), and preferred less when the adjective or participle is predicative (as when used after a form of to be): a common style is to write "a well-prepared student" but "The student is well prepared". However, it is also common to use a space both ways (as phoog prefers). It can even be possible to hyphenate in predicative as well as attributive position: see the following questions: