Is the prayer of one afflicted with urinary incontinence valid if he does not perform ablution for the 'Asr prayer, while maintaining the ablution he performed for the Dhuhr prayer, and the cloth he places to prevent drops from falling off? And in this case, would he be considered as having attained the virtue of maintaining the opening Takbir for forty days?

A person afflicted with سلس (urinary incontinence) is someone who does not find a regular time during which the flow of urine stops, sufficient for them to perform their purification (Wudu') and prayer. However, if the flow stops at a known time, then they must delay their Wudu' and prayer until that time.

A person afflicted with سلس must perform Wudu' after the time for each prayer begins, but this is contingent upon something having been discharged. However, if nothing has been discharged since the time of their Wudu', then it is permissible for them to pray the subsequent prayer with that same Wudu'.

If no urine has been discharged from you, and no other nullifier has occurred, then your 'Asr prayer performed with that Wudu' is valid. But if urine has been discharged from you, then your 'Asr prayer is invalid, and you must repeat it now.