"I will work at the university starting April 1st."
It is common to use arabic numerals for expression of date. "で-つとめます" is strange. "に-つとめます", or "で-はたらきます" is correct. I think that "はたらきます" should also be included in one of correct solution because it is correct.
Agreed. Not only is this usage strange, it is considered incorrect by pretty much every Japanese textbook and teacher. に+つとめる is considered the only 'correct' option. で+つとめる would just be plain wrong and should not be offered as the default translation.
As far as the Arabic numerals, while I do concede your point, it is important at this stage of learning to require Kanji numerals.
Okay, thusfar I've learnt しごとします, はたらきます and つとめます for "to work" (the ます-forms at least). Can they be used interchangeably or are there subtle differences?
仕事①をする would be more akin to 'do work; perform tasks'. 仕事 can be counted, as in 「沢山②の仕事があります。」, which would normally be translated as 'a lot of work' but also as 'many tasks/jobs'. 働く③ would be closer to the English for 'to work; to put your energy into something. 働く, as a verb, is non-countable.
もっと仕事をしろ！ would be 'Do more work!'
もっと働け！ would be 'Work more/harder!'
As for 勤④める, it means to 'be employed at/with'. 務める and 努める are also readings for つとめる, so it would be helpful if you specify which you are asking about.
①しごと ②たくさん ③はたら(く) ④つと(める)
仕事 means job or work (noun). It is actually the best translation for the sentence above.
働き means any kind of work, whether it's for an employer or for yourself. It means physical labor.
勤め refers to employment/hiring. It means you have joined an organization.
The correct translation of the above statement is not what is given, but is: 四月一日大学に仕事始まります。