It’s about seven tenths of a mile from St Paul’s Cathedral to the Tudor Gateway in Chancery Lane, the main access point for Lincoln’s Inn in 1623.
The Inn’s website tells us that the great oak doors date from 1564. The Gate House itself was built during the years 1517 to 1521, with bricks dug and made within the Inn. Above the gateway, facing eastwards into Chancery Lane, there are the arms (from left to right) of Henry de Lacy, Earl of Lincoln; Henry VIII; and Sir Thomas Lovell.
Donne certainly could have walked over from the Deanery in the southwest corner of Paul’s Churchyard, but I suspect he rode a horse, or, perhaps more likely, rode in a carriage with George Montaigne, Bishop of London, and other members of the Bishop’s entourage.