Learn Icelandic: Þetta er yfirmaðurinn sem hún vinnur með.

Just like in English ('the boss she works with'), Icelandic puts the preposition at the end of a relative clause. When you use the relative word sem (that/who), the preposition must go at the end. You cannot say með sem for 'with whom'.

Does the preposition með change the case of the noun yfirmaðurinn here?

No, it doesn't. The word yfirmaðurinn is in the nominative case because it goes with Þetta er (This is) in the main part of the sentence. The connecting word sem never changes case, so the preposition með just sits at the end of its clause without affecting the main noun.

Why do we use the neuter word Þetta when yfirmaðurinn is a masculine noun?

When you are introducing someone or something by saying 'This is...', Icelandic always uses the neuter form Þetta er.... You do not match the pronoun's gender to the noun until you refer back to that person later in the conversation.

What are the parts that make up the word yfirmaðurinn?

It is a compound word. yfir means over or above, and maður means person or man. So an yfirmaður is an 'over-person', which is how you say boss or manager. The -inn at the very end is the attached word for 'the'.