Most tract table data files from NHGIS include metro area codes. You can determine which field contains that code by reading the “codebook” text file that accompanies the data file. (E.g., in the most recent files, the metro area code is given in a field labeled “CBSAA”.)
NHGIS shapefiles and time series tables for tracts don’t generally include a metro area code. To get metro area codes associated with tracts in those files, you could join them to a standard “source table” file for tracts, keying on the common “GISJOIN” identifier that appears in all NHGIS files.
If you need a lookup from metro area code to metro area name, you could use a source table for metro areas (i.e., core-based statistical areas, or CBSAs), which contains both the codes and the names.
For more information about MSA definitions and delineations, and how they’ve changed over time, see this page on the Census Bureau website.