Come what come may
This could be rephrased in more modern grammar as "Let what may come, come!"
Shakespeare's first "come" is a subjunctive form. The way it is used here is similar to how it is used in the phrase "be it enacted" (listed on Wikipedia as an example of an archaic English subjunctive), which has the meaning "Let it be enacted," or the phrase "God save the queen," which means "Let God save the queen" or "May God save the queen." The English subjunctive does not have distinct forms for different grammatical numbers or persons.
"What come may" is an archaic way of saying "what may come."
As Anonym mentions in a comment, some set phrases that use the subjunctive like this may still be used in modern English; we can say "Come what may" with the same meaning. Here, the initial "come" is subjunctive, and the auxiliary verb is used by itself, with the second "come" left implied.
Time and the hour runs through the roughest day.
There are several reasons why Shakespeare may have used "runs" rather than "run" here. One is that, as other people have mentioned, we can occasionally use a singular verb after two singular nouns that are connected by a conjunction, but considered one unit (such as "bow and arrow"). To some extent, this is true today. However, you should also remember that Shakespeare used archaic grammar, and there are also many cases where he used a singular verb with subjects that would take plural agreement in today's English. For examples and discussion of this (and other grammatical oddities of the Bard), see this page on Shakespeare's Grammar, or this page by Alan Powers.