"He was in good form and happy to be spending time with his wife and son," the Cardinal (who met up with the Russert clan in Rome) tells PEOPLE. The Meet the Press host and his wife, Vanity Fair journalist Maureen Orth, were celebrating son Luke's recent graduation from Boston College.
Foley – who has known the devout-Catholic Russert for 24 years and even baptized Luke – accompanied the trio to Saint Peter's Square, where they had front-row seats for an audience with the Pope.
"He was a very religious person," Foley says. "He never tried to hide his faith but never forced it on anybody either."
Russert – who died of a heart attack June 13 in Washington, D.C., soon after returning from Italy – met Pope Benedict XVI last April, Foley adds. "But Tim thought it would be nice for his wife and son to experience [it], too. "
After the Papal audience, Foley took the family to a "small restaurant near the Vatican," where he and Russert ordered a filet steak and talked about politics.
"It was a very hot day and both Tim and his son were very hot, because they are both big men," he says. "We talked about his work and the presidential campaign ... This was his year."
After lunch, Foley arranged for his driver to take the Russerts to see the Sistine Chapel "through a special door," which allowed them to skip the long lines outside the Vatican museum.
Foley says Russert had already seen the Michelangelo frescoes many times before but wanted Luke and Maureen to see them as well. "He was very devoted to his son," adds the Cardinal, "and obviously very proud of him."
Funeral services for the late NBC star will be held on Tuesday and Wednesday.
For the in-depth story on Russert's shocking death, pick up a new issue of PEOPLE on newsstands Friday