Learn Icelandic: Ég vinn á mánudaginn.

It is in the accusative case. When we talk about a specific, single day like this upcoming Monday, we use á plus the accusative form. If you were talking about a repeated habit (working every Monday), you would use the dative case instead.

Why is there an -inn at the end of mánudaginn?

The -inn is the attached word for "the" (the definite article). In Icelandic, to say you are doing something this coming Monday, you literally say "on the Monday".

What would the sentence look like if I wanted to say "I work on Mondays" as a regular habit?

To talk about a routine, you switch to the dative plural case. You would say Ég vinn á mánudögum. Notice how the ending changes to -um, which is the classic sign of the dative plural, and the attached "the" is dropped!

Can ég vinn mean "I will work"?

Yes! Icelandic doesn't have a specific future tense form of the verb. Instead, it frequently uses the present tense (vinn) paired with a time phrase (á mánudaginn) to make it clear you are talking about the future.