Learn Icelandic: Iðnaðarmaðurinn ætlar að byggja húsið sjálfur. - The tradesman is going to build the house himself.

Because it is emphasizing the subject of the sentence, Iðnaðarmaðurinn (the tradesman). As an intensifying pronoun, sjálfur has to agree in gender, number, and case with the noun it refers to. Since Iðnaðarmaðurinn is masculine, singular, and in the nominative case, sjálfur must be too.

Could sjálfur go right after the tradesman instead of at the end of the sentence?

Yes! You can say Iðnaðarmaðurinn sjálfur ætlar að byggja húsið (The tradesman himself intends to build the house). Placing it right after the noun makes the emphasis a bit stronger, but putting it at the end of the sentence is also very common and natural in Icelandic.

Why doesn't húsið affect the form of sjálfur since it is right next to it?

Because sjálfur is emphasizing the tradesman, not the house. If it were emphasizing the house (for example, if he were building the house itself, rather than an extension), it would need to match húsið and be in the neuter accusative form (sjálft). Since it means he is doing it himself, it locks onto the subject instead.

What case is húsið in here?

It is in the accusative case because it is the direct object of the verb byggja (to build). However, because hús is a neuter noun, its accusative form looks exactly the same as its nominative form.