In essence, you're asking for how possessive pronouns preceding an object of a verb pair with the subjects in a neither...nor construction, especially when that object could pertain to either subject. In other words, what concord is there between a pronoun and its noun phrases in the case of neither...nor? The options break down like so:
Does the ...
"Neither my friend nor I feel their best in the morning."
You are talking about yourself almost as a third person, so can use 'their', which is definitely appropriate for my friend.
Look at this article on singular they: https://en.wikipedia.org/wiki/Singular_they