Charlottesville, VA is an example of how the new MET2013 protocol for identifying metropolitan areas differs from the protocol used in METAREA. Under the METAREA protocol, Charlottesville would have been considered an identifiable MSA because, as you pointed out, PUMA 51090 lies entirely within a single metro area, and thus has 0% commission (no persons residing outside a metro area are identified as living in it). However, MET2013 takes both commission and omission (where a MET2013 code is not assigned to some residents of the corresponding MSA). So, while you can confidently state that every person living in PUMA 51090 is also living in the Charlottesville MSA, the sizable portion of the Charlottesville MSA that lies in other PUMAs (which themselves include areas outside of the MSA) means that a frequency or analysis of MSA would not accurately represent the Charlottesville population.
If you are willing to accept the omission error, you can use the Crosswalk between 2013 MSAs and 2010 PUMAs spreadsheet available at the bottom of the MET2013 description page to identify PUMAs with 100% of their population within a single MSA.
I hope this helps.