You can't say "Around/About under $700 a month". You can't byse them with 'under'. This is because 'around' and 'about' mean 'more or less'. They both mean close to $700 but perhaps a little more or less. It doesn't make sense to say 'a little more or less than under $700 a month'.

Yes, you can say those. The mathematical meaning of "about under" or "around under" is fuzzy. I would not want to try to define what they mean exactly. However, there is no "Preposition Police" to enforce any rules about which prepositions can or cannot be used. In your sentence, I would interpret "about" or "around" to mean that you are uncertain about what you are about to say. The sentence would thus mean "I'm not sure about this, but roughly I think the budget will be under $700 a month" So you can use those words to produce a valid sentence whose fuzziness reflects your own uncertainty.

yes sounds good,

about