When possible, I believe they call ahead and arrange for a Red Elephant Club member, A-club member, or other 'friend of the University' to pick him up and transport him around. If that's not feasible, they arrange to have a rental car or taxi waiting at the airport.
As for recruits flying on University owned planes, no. A couple of years ago, the NCAA passed what is commonly called the Willie Williams rule which outlawed Universities using extravagant transportation (private planes, helicopters, stretch limos, exotic cars, etc.) to transport recruits.
Prior to that rule, University owned/leased/hired jets would fly recruits to and from T-town for official visits. Now, the prospects either fly commercial airlines (usually to B'ham where the University picks them up) or furnish their own transportation with the University reimbursing them for mileage.
