You're close. Some issues:
Since "his army" is refering to the subject of the sentence (Caesar), you need to use a form of suus, a, um for "his." "eius" is a pronoun and would refer to another person's army (not Caesar's).
You were asked to use "qui + subjunctive," but you've changed qui to quem. I understand why you thought of doing this (to agree with the accusative case of the referent earlier in the sentence). However, relative pronouns (qui, quae, quod) must agree with their referent only in number and gender, NOT case. You must use "qui" because it is the subject of that clause (of the verb obsideret)--the subject could not be in the accusative case.
1) Should exercitum be in the accusative or the genitive (as a partitive)? ie. should it be "partem eius exercitus"?
Partitive genitive is correct. In the accusative you'd have two nouns as the object and it wouldn't make sense.
2) Am I correct in using the singular form of the relative pronoun and the subjunctive verb - an army is multiple people so should it be a 3rd plural pronoun and verb?
The singular form is correct, because you are referring to a noun that is in the singular. Don't worry about whether you are talking about a group of people or not. As I explain above, the relative pronoun will refer to a noun earlier in the sentence, and whatever that noun is will determine the number of the pronoun.