Only a small part of University Ave is VA 331 and it is not posted in any kind of route marker (shield nor rectangle).
Here is the south end:
https://goo.gl/maps/x1JuZBdNYPR2Here is the north end:
https://goo.gl/maps/jEv81taExF72Total length is 0.6 miles
It is not surprising it is not posted (nor are any of the pieces on JMU campus between I-81 and US 11) given that VA 280 and VA 253 are not posted within the City of Harrisonburg, with the exception of I-81 BGSs at VA 253.
I have had the thought that perhaps a separate set of files for Virginia could be created for the routes that are posted only with a rectangle (currently not in the state highway set in TM) or no longer posted at all but are still verifiably still in the system, such as your VA 331 question above. Maybe call it unposted primary routes or similar as a lower tiered system.
Doing this would get these routes into TM (I'm sure I'm forgetting a couple):
113, 192, 209, 213, 313, 317, 324, 327, 331 Univ Dr, 332, 335, 339, 342, 345, 350, 355, 366, 367, 375, 377, 382, 390, 393, 394, 399, 420
Parts of 302 and 303 would also be added under this concept...