Is it possible to pray on time without the need for making up the prayers (qada'), despite the release of drops of urine after urinating due to an enlarged prostate, and the inability to shower or change clothes every time?

The discharge of urine drops necessitates ablution (wudu) and washing only the area of the body and clothing affected by the urine. In the case of urinary incontinence (sals al-bawl), one must wash the private parts and any clothing stained by urine, place a barrier to prevent the spread of urine, then perform ablution for each prayer, and pray in that state. Whatever exits thereafter does not matter, and your prayer is valid. However, if the discharge of urine stops after a short period, one can go to the restroom a sufficient time before prayer to cleanse oneself (istinja (definition)) and perform ablution, then pray.