I have two questions,
1. Who is the companion of whom the Prophet Muhammed (Sallallahu Alaihi wa Sallam) said that if there would be a Prophet after me, it would be that companion? Is the companion Abu Bakr or Umar?2. When the Mosque of Madina was being founded, did our beloved Prophet describe the succession of Caliphate, that Abu Bakr would be the first then Umar, Usman and Ali? Is there any evidence proving this?

All perfect praise be to Allaah, The Lord of the Worlds.  I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.

The Companion whom you are asking about is ‘Umar ibn Al-Khattaab . At-Tirmithi and Ahmad both, reported that ‘Uqbah ibn ‘Aamir narrated that the Prophet, sallallaahu ‘alayhi wa sallam, said: “If a prophet were to come after me, it would be ‘Umar.”

As for the sequence of the caliphate of the four Caliphs, there is no Hadeeth (narration) of the Prophet, sallallaahu ‘alayhi wa sallam, regarding this. However, there were some indications about the caliphate of Abu Bakr and ‘Umar .

Al-Bukhari and Muslim both, reported the Hadeeth of Saalim from his father about the dream of the Prophet, sallallaahu ‘alayhi wa sallam, about Abu Bakr and ‘Umar both. The Prophet, sallallaahu ‘alayhi wa sallam, said: “While I was sleeping, I saw myself standing at a well; on it, there was a bucket. I drew water from the well as much as Allaah wished. Then Ibn Abi Quhaafah (i.e. Abu Bakr) took the bucket from me and brought out one or two buckets (of water) and there was weakness in his drawing the water, may Allaah forgive his weakness for him. Then the bucket turned into a very big one and Ibn Al-Khattaab took it over and I had never seen such a mighty person amongst the people as him in performing such hard work, till the people drank to their satisfaction and watered their camels that knelt down there.”

Imaam An-Nawawi said when commenting on this Hadeeth, “That is an indication about the caliphate of Abu Bakr and ‘Umar both, and that their reign is correct and was described; the Muslims accordingly benefited from it.”

Allaah Knows best.