Is the prayer attained by attaining the opening takbir or less than one rak'ah, and is re-performing the prayer obligatory for someone who established the prayer two minutes before the entry of the Asr time due to lack of time, and is his action considered a legitimate excuse?

Delaying a prayer until its time expires is religiously forbidden. You are not obligated to repeat the prayers you have already performed, as they are valid whether performed within their prescribed time or as a make-up (qada'). The Hanbalis hold that a prayer is considered to have been performed on time if the opening takbir (takbirat al-ihram) is uttered within its prescribed time, and it does not become invalid if the time expires while one is performing it, even if the delay was intentional. The majority of jurists believe that a prayer is considered to have been performed on time if a complete rak'ah, with its two prostrations, is performed within the time. Some of them hold that it is considered performed if only the bowing (ruku') is done within the time. Some Hanafis and Shafi'is believe that whatever part of the prayer is performed within its time is considered on-time performance (ada'), and whatever is performed after its time is considered make-up (qada').