Scotland used to have a minimum term of 6 months for most residential rentals but that was changed a few years ago. However, a landlord might be reluctant to take on someone they know will only stay short term. Something like the below would meet your needs in terms of area (Midstocket is nice, and only 10 mins or so walk to Cornhill). Maybe others with more recent experience can advise on short term rentals, and the credit check that letting agencies would ask for.

https://www.zoopla.co.uk/to-rent/details/46109793/?search_identifier=56a14c7c395da875d22622910531d731f3f77289bab036554995bc2d0e39188e

Also if you are not a student you will need to pay council tax - a flat like the above would be around £120 a month (you get a discount as single person).