I looked for its meaning and I found this from The Free Dictionary:
A large, sudden, and widespread demand for something.
I am not sure this is the right answer for the next context:
No French sovereign could dare to withdraw without avenging the first check met with by the French army since Waterloo, and thus was the Emperor rushed on to fulfil his own destiny.
Maximilian in Mexico - Woman's Reminiscences of the French Intervention 1862-1867 Sara Yorke Stevenson
The definition that you found is for the noun "rush", which can be used with the preposition "on". However, in your example "rush" is not a noun; the preceding instance of "was" and the "-ed" suffix suggest that it is instead the past participle form of a verb. (You may want to review how to form the passive voice.) Merriam-Webster's definitions aren't a perfect match for this meaning, but I'd say that the closest is:
to push or impel on or forward with speed, impetuosity, or violence
The closest definition for "on" is:
2a: forward or at a more advanced point in space or time
| went on home
| later on
I doubt that the emperor was rushed forward literally "at a more advanced point in space or time"; the use of "on" here is somewhat figurative and means something more like "forward toward some goal".
EDIT:
Another possibility for "on" (suggested in a comment below) is:
b: in continuance or succession
| rambled on
| and so on
