Old Nassau is an old timer (1862) but no, Princeton and "Old North" is more than a whoop and a holler away.
I have found "Old Amherst," Old Brown, Old Harvard, Old Hobart, Old Williams, Old Yale, (too many " " to type) and probably others.

I suspect that it may have been a parody of "It's a Way We Have at Old Harvard," because that is cited as the tune for "Jubilate" in "Carmina Collegensia" (1868), while "Old Dartmouth" is cited as the "Jubilate" tune in "College Songs" (1860). However, "It's a Way We Have at Old Harvard" is printed with music in "College Songs."

"It's a Way We Have at Old Princeton" appears in "Carmina Princetonia," so "It's a Way ..." seems to have been used at several of the old schools.

Lyr. Add: JUBILATE
AI Baker Thompson
Air: Old Dartmouth

We have come together to-night, Boys,
With hearts merry and light, Boys,
In accordance with our right, Boys
To have a jubilee, etc.
[To have a jubilee,
To have a jubilee,
To have a jubilee,
In accordance with our right, Boys,
To have a jubilee.]

Released from care and vexation,
And the pangs of recitation,
We're just in a situation
To have a jubilee, etc.

We've studied mathematical science,
In sulLen, reluctant compliance,
With "the laws" which we set at defiance
To have a jubilee, etc.

We loathe mathematicas artes,
Thesis et ictus et arsis,
In animo all of our class is,
To have a jubilee, etc.

Then fill up the bowl to the brim, Boys,
With brandy, nor wine, nor gin, Boys,
For these cause the brain to swim, Boys,
Hurrah for a jubilee, etc.

The lyrics of "Jubilate" from "College Songs," 1860, compiled and arranged by C. Wistar Stevens, pub. Russell and Tolman, Boston.