I'm afraid I can only touch on this in general terms as the history of Hajj practices is not something I'm terribly familiar with. Basically you would likely have taken one of a few routes. The two main Hajj caravan routes in the Middle Ages and up until the 20th century would have set off from Damascus and Cairo. Depending on where our hypothetical traveller was in Anatolia it might have been significantly easier and quicker to sail to Cairo, rather than make the trek to Damascus. In any event you're going to have to then set off on a rather arduous overland journey with hundreds if not thousands of other pilgrims in the annual Hajj caravan. You'll be stopping at a number of pre-set oases and waypoints, but it's not until the Ottomans takeover the levant that this really gains adequate infrastructure and support. It's going to be pretty uncomfortable and potentially dangerous. As far aware there wouldn't be any real issues crossing the border lands from Ottoman into Mamluk territory, other than the generalized threat of banditry along the entire route.

In both instances you'll have the chance, at Suez (if you came from Cairo) or at Aqaba (if you came from Damascus) to significantly shorten the trip by taking a ship down the Red Sea to Jeddah.

As for how the experience of Hajj practices would differ from today, in Mecca itself you'd have pretty bare-bones infrastructure to support pilgrims. Major renovations to the "grand mosque" would be undertaken by the Ottomans and then again in the 20th century by the Saudi Bin Laden group which more or less created the Grand Mosque as it can now be seen. That said, there would have been far fewer pilgrims than today. Today there are a few million pilgrims annually. I'm struggling to find the statistic but IIRC the number up until the early 20th century would have been in the low tens of thousands, and many of those would have been "locals" from Arabia itself.

These essays on the Hajj put together by the British Museum might be of interest, I've only just skimmed them to confirm some of what I've written here but they gel with what I've come across in the past and include essays from several scholars I'm familiar with and would trust: https://www.britishmuseum.org/PDF/The_Hajj_collected_essays_BM.pdf