Can a bladder infection in a woman lead to a man getting prostatitis, or the reverse?

Not as a rule: Though the types of bacteria that cause utis in women and prostatitis in men are similar, this is generally not possible. However, some men will develop prostate infections related to stis like chlamydia or gonorrhea, and vice versa for women who might have urethral infections. Safer sex practices and regular STI testing is always important.

Not likely: Assuming we're not talking about sexually transmitted diseases, no this is not a concern. No transmission of these gram negative organisms in either direction.