While traveling with Tyrion, Jorah was infected with greyscale, a deadly disease that Daenerys sent him to find a cure for during Season 6. Fans were thus surprised when Jorah, or at least his infected hand, appeared in the Season 7 premiere at the Citadel. So how did Jorah get to the Citadel on Game of Thrones? It's a bit of a mystery.
Back in Season 4, Daenerys learned that Jorah was once a spy who reported all of her comings and goings to Robert Baratheon. Because of this betrayal she banished him, threatening that if he did not leave she would have his head thrown into Slaver's Bay. Still, when Jorah teamed up with Daario to save her from the Dothraki (not that she needed saving), she was hesitant to banish him again. However, the choice was made for her when he revealed his greyscale.
After Jorah finally revealed he was in love with Daenerys, she commanded him to find a cure so he could be by her side when she took the Iron Throne. It seems Jorah didn't waste any time and went straight to work looking for a cure. While the journey to the Citadel was probably a long one, he did have a bit of head start on Dany, since she first had to return to Meereen to clean up the mess Tyrion made with the slave masters. This would explain how he beat her to Westeros.
Still, how did he travel? Jorah had nothing on him except his weapons and his greyscale. Plus, he's clearly sick. How far did he get before the greyscale got to the level that it is now? Did he even make to the Citadel on his own or did someone find him and bring him there?
Unfortunately, there aren't any real answers just yet, but hopefully Jorah was able to do what Daenerys ordered him to do and found a cure. However, based on the look of his arm, that seems unlikely. But regardless of whether or not Jorah has found the cure yet, fans seem confident Sam will.
Reddit user StormSS was able to zoom in on one of the books Sam was reading about dragonglass. Though it's difficult to make out what it says exactly, it seems to suggest that Dragonglass may be a cure for greyscale. Additionally, other users commented how Shireen survived her case of Greyscale and maybe that was because she grew up at Dragonstone, where an underground mountain of dragonglass resides.
If this is true, Daenerys may actually hold the key to Jorah's survival and not even know it. Talk about a tragic form of irony. Hopefully, if Daenerys really does have the cure, Sam can make the connection quickly enough to save Jorah's life. This is Game of Thrones, though, and it wouldn't be surprising if Daenerys got there just a little too late.