It is said that menstruation takes place after 14 days of ovulation. So if the ovum dies in 24 hrs, then will it stay for another 13 days in the fallopian tube/uterus for the menstruation to start?

Yes, it stays for 13 days and is still not shed until the blood capillaries and uterus linings start shedding. In simple words, The ovum waits for the lining to shed and the uterus linings wait for fertilisation of the ovum.