Translation:I did not work the day before yesterday.
It's roughly "As for the day before yesterday, I did no work". Think of whatever is before the "は" as the subject line of an email, or the bold line (possibly starting with Re:) at the top of a modern-style business letter. In fact, "Re:" is a good enough translation of the "は", but applies to what's before it.
i think it's correct, but then you change a name for a verb. shigoto = job; it means work, but it's a "name", a substantive. hataraku = work; it's a verb.
when you say "shigoto shimasendeshita" (しごとをしませんでした), you're conjugating the verb suru, that means "to do". It's like "the job i didn't do".
when you say "hatarakimasendeshita" (はたらきませんでした), you're conjugating the verb hataraku. It's like "i didn't work".
If I get it right, that's it, but please anyone correct me if you find something wrong :)
There's nothing wrong with your word order. You can usually put time phrases wherever. Although it's a little strange-sounding because dropping "the" at the beginning is informal, but "did not" is formal compared to "didn't".
... Or are you a native speaker and you're just asking how Duo sees it ...
You don't put 2 'wa' particles. In this case, you would use 'ga' が after おととい. This is due to が also being a topic particle, however it is considered a 'lesser' topic particle than は, i.e. the major topic is は and the secondary topic is が . So the full sentence would be 私は一昨日が仕事をしませんでした (わたしはおとといがしごとをしませんでした)
Nope, しごと is a noun, so you can't just tack on the ませんでした verb ending to it.
In the sentence above, the verb is actually しませんでした, which is the negative past tense form of the verb "to do". So technically, if we were being strict, this sentence should actually have been translated as "I did not do work the day before yesterday"
If you want, you can use the verb for work, はららく（働く） So it would be おとといは はたらきませんでした
You don't need は, but it can emphasize おととい and has a slightly different nuance. With the particle, the sentence is more like "As for the day before yesterday, (I) did not do work/job." Without it, the meaning is more like "The day before yesterday, (I) did not do work/job." The particle specifies that I, the subject, did not do work on the specific day, the day before yesterday.