There is much wisdom as to why Prophet Muhammad ﷺ said those words. First of all to simply answer your question if this hadith is sound or no then yes it is sound and recorded in authentic books of hadith. Messenger of Allah ﷺ said: "If there was to have a Prophet after me, it would have been 'Umar bin Al-Khattab." Secondly what the Prophet ﷺ said does not mean that he considers Umar (r.a) a prophet. What he says is based on the fact observed and agreed upon by all companions which is answered in another hadith where Messenger of Allah ﷺ said: “Indeed Allah has put the truth upon the tongue and in the heart of Umar." And Ibn 'Umar said “No affair occurred among the people, except that they said something about it, and 'Umar said something about it, except that the Qur'an was revealed in line with what 'Umar had said.”
A prophet speaks nothing but the truth whether from his ownself or directly from Allah ﷻ, and Umar (r.a) was very strong in this prophetic characteristic, not that he was a prophet or considered one. The reason how we find that truth was placed on the tongue of Umar (r.a) and in his heart was because there were several instances when Prophet ﷺ and his companions faced with situation where they awaited Allah’s response through revelation and Umar (r.a) shared his opinion. Later the verses revealed confirmed the opinion of Umar (r.a). Below commandments revealed in Quran corroborated Umar’s opinion:
1. Umar (r.a) suggested that the station of Abraham in Makkah should be used as a place of prayer. Later an injunction to this effect was revealed to the Holy Prophet in Surah Baqara (chapter 2).
2. Umar (r.a) suggested that the wives of the Prophet ﷺ should be veiled. Later a verse was revealed enjoining the wives of the Prophet ﷺ to be veiled in Surah Al Ahzaab (Chapter 33).
3. Umar (r.a) suggested that the use of wine should be prohibited. Thereafter Allah enjoined the prohibition of wine in Surah Al Maidah (Chapter 5).
4. Abdullah bin Ubbay was one of the hyprocites, insincere in his professions and was the enemy of Allah and Prophet Muhammad ﷺ. When he died the Prophet led his funeral prayer. Umar (r.a) suggested that the Prophet ﷺ should not pray at the funeral of those who were the enemies of Allah and the Prophet. A verse was later revealed enjoining the Prophet not to pray at the funeral of those who were the enemies of Allah and His Prophet ﷺ in Surah Taubah (Chapter 9).
5. When there was an imputation against the conduct of Prophet's beloved wife Ayesha (r.a), Umar (r.a) said that this was a grievous calumny. Later a verse was revealed declaring the episode as a calumny and establishing Ayesha's innocence in Surah Nur (Chapter 24).
6. When the Adhaan was originally proposed the contents of the call were: "I testify that there is no god but Allah-come towards prayers." Umar (r.a) suggested, "The words 'I testify that Muhammad is the Messenger of God' should be added". A revelation corroborated this suggestion.
7. Once a Jew said to Umar (r.a): "Verily Angel Gabriel who speaks to your Master is our enemy." Umar (r.a) retorted, "Whosoever is an enemy to Allah, or His angels, or His Messengers, or Gabriel, or Michael, verily Allah is an enemy to the unbelievers." Later a verse was revealed declaring that Allah was the enemy of unbelievers in Surah Baqara (Chapter 2nd).
These instances confirms the truth that Allah ﷻ placed on Umar’s tongue and in his heart just as the Prophet ﷺ truthfully said about him. Now if the question arises how is this possible then it is very much possible for a believer to have insight in to what is right and wrong and this comes due to light of faith and conviction just as the Prophet ﷺ taught us: "Beware of the believer's intuition, for indeed he sees with Allah's Light." The intuition occurs in a believer’s heart and he confirms it with his or her tongue. The knowledge of this intuition occurs in our souls whose origin is heavenly and if pure gains access to the presence of angels. In the Quran it is said He has knowledge of everything (Kulli Shayyi’n) and He compares His Knowledge to Nur (Light): (Al Quran 24:35) "….God guides whoever He will to his Light; God draws such comparisons for people; God has full knowledge of everything" Such was the case of Umar (r.a) except that he had a very strong intuitive ability. In fact the Quran also speaks about Khidr (alaihi salaam) who was given mystical insight from Allah ﷻ though he was not a Prophet: (Al Quran 18:65) ".. on whom We had bestowed grace from Ourselves and to whom We had imparted knowledge [issuing] from Ourselves." But what is important to note is that not only him but several other companions of the Prophet also had knowledge and insight through intuition. Please read my answer to this question that explains how Allah (swt) reveals the knowledge of unseen to his chosen ones who are not Prophets & Messengers. This also includes how Adhaan was revealed to Umar (r.a) via a dream and dreams are also means through which Allah (swt) reveals the knowledge of unseen. Zaid Shaw's answer to Does only Allah have knowledge of the future?
 Jami Tirmidhi - Vol. 1, Book 46, Hadith 3686
 Jami Tirmidhi - Vol. 1, Book 46, Hadith 3682
 Professor Masud-ul-Hasan,1982, Islamic Publications, Lahore, Pakistan - http://www.alim.org/library/biography/khalifa/content/KUM/20/12
 Jami Tirmidhi - Vol. 5, Book 44, Hadith 3127