Is it possible for someone from our era to be alongside the Messenger (PBUH) in Paradise, or alongside his Companions in rank, and what are the deeds that lead an individual to those stations?

The prophets hold the highest ranks in Paradise. The Companions feared that the exalted status of Prophet Muhammad, peace and blessings be upon him, might prevent them from seeing him and being near him. So, one of them came and asked him about this, and Allah Almighty revealed: (And whoever obeys Allah and the Messenger - those will be with the ones upon whom Allah has bestowed favor of the prophets, the steadfast affirmers of truth, the martyrs and the righteous. And excellent are those as companions). This means they will be with them in the same abode and enjoy the same bliss. Whoever obeys Allah and His Messenger will, by Allah's grace, be a companion to those upon whom Allah has bestowed favor. It is mentioned in Sahih Muslim that the one who sponsors an orphan will be with the Prophet in Paradise, and the Prophet said to Rabia'a ibn Malik al-Aslami when he asked for his companionship in Paradise: "Then help me against yourself by prostrating much." And it is in Al-Bukhari and Muslim that whoever loves Allah and His Messenger will be with those he loves. Thus, sponsoring an orphan, frequent prostration, loving the Prophet and his Companions, and performing many good deeds elevate their doer to the companionship of those upon whom Allah has bestowed favor.