No, she never beat him through all 5-6 years they're in school. Earning her the nickname of "Eternal 2nd" (much like a certain Special A heroine)
Even the one time she got first place was in an event where male and female students are separated, thus without him being the blockade she got 1st place...for female group.
In fact her desire to beat him is so strong that it's only when he himself ask her what's her plan after graduation, on graduation day, that she finally mention she want to be receptionist, to the surprise of everyone.
That said, she's...oblivious to some details...I'll paraphrase a certain moment that capture it best.
Nana "Teacher, how can I improve my grade?"
Teacher "...You already got perfect score tho?"
Nana "I still can't beat that guy"
If a test is grade out of 100, she'd get 100, and he'd somehow get 103.
And nope, 0 chance of romance, this girl is too dense/workaholic for that.