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.
If that rich man sets a condition that those who go for Hajj or ‘Umrah with him must be among those who cannot afford the costs of Hajj or ‘Umrah, you will not be allowed to go with him as long as you are able to cover the costs. If he does not put this condition on those who go with him, the expenses of Hajj in this case will be like a gift from him and you are allowed to accept it without harm. However, it is preferable to perform Hajj from your own money in order to achieve the full reward. It should be noted that performing Hajj and ‘Umrah result in the increase in one's sustenance, not the opposite. The Prophet, sallallaahu ‘alayhi wa sallam, said: "Perform Hajj and ‘Umrah repeatedly for they expel poverty and sins as the bellows purify the impurities of iron." [At-Tirmithi]
Allaah Knows best.