I think its possible to do well in a masters of mechanical engineering program if you either take a couple of post bacc undergrad courses in mechanical engineering or pick up some quality textbooks, read through them, and solve an abundance of problems prior to enrollment.
You should already have a strong understanding of mathematics, Newtonian mechanics, and thermodynamics. Your critical thinking skills should be developed on par with other engineers. Depending on what your focus is in the mechanical engineering program, you will need to cover the lost ground that other engineers learned in undergrad.
For example, if you are choosing a thermodynamics/fluids program, buy introductory books on thermodynamics, heat transfer, and fluid dynamics. You can get feedback on forums for recommended texts, contact the potential graduate professors for what they would recommend, or you can look at the required texts for the undergrad engineering program that offers the masters program and pick from there. Read through them. Attempt as many problems as you can. Get your hands on robust solution manuals.
Undergrad for mechanical engineering covers a lot of breadth and not so much depth. This can be to your benefit because most masters programs are focused on depth and not breadth. You can save a lot of time by not learning the undergrad courses that are not necessarily applicable to your focus. I would say here is the breadth of areas covered in my undergrad:
1. The mechanical design path: Strength of Materials -> Mechanical Design 1 -> Mechanical Design 2 & Finite Element Methods 1 -> Mechanical Design 3
2. The thermofluidics path: Thermodynamics 1 & Fluid Mechanics 1 -> Thermodynamics 2 & Fluid Mechanics 2 -> Heat Transfer
3. Dynamics & Controls path: Advanced Dynamics -> Vibrations -> Control Systems
4. Mechantronics path: Microcontrollers & PLCs & Several Programming Courses -> Mechatronics Design 1, 2, & 3
Do you already have a job or knowledge to what you would like to focus on within mechanical engineering? If not, I would get the undergrad degree because it is more flexible. It would be a shame if you focused of thermofluids and found out later that you would rather do control systems or design structures & mechanisms.