Ceiling Fan Junction Box Converter/Bracket 1/2 inch width difference
If your box is that small it is almost certain that it is not a fan rated box. If it is not, then the box MUST be replaced with a fan rated box.

Sorry, but even with approval, someone unlicensed, uninsured, and unqualified SHOULD NOT be doing electrical work in a place that they rent.
This is a perfect example; something as "simple" as changing a ceiling fan turns into a project since the box has to be replaced, and in an older house this can be met with issues.