"dran" or "daran" usually references some subject, state or location which was mentioned in a previous sentence. For example:
"Michael hat seinen Termin verpasst. Daran bin nur ich schuld."
"Michael missed his appointment. That is completely my fault."
"Täglich wird es wärmer. Daran erkennt man, dass der Frühling kommt."
"It is getting warmer every day. This is how you can tell that spring is coming."
"Die Tür ist wackelig. Lehne dich bitte nicht daran an."
"The door is wobbly. Please don't lean against it."
Now, for the sentence "Du bist dran." there is no direct noun that is being referenced here. But since this is somewhat of a set phrase, "dran" implicitly refers to "die Reihe"/"the queue" or "the line" (either a physical or metaphorical one). So actually it means
"Du bist an der Reihe"
"You're next (in line)"
as you correctly translated.
Moving to your actual question, let's look at the problematic sentence again:
"Meistens bin ich mit der Vorbereitung der Prüfung spät dran und kann mir dann nicht mehr alles merken."
"Usually I am late for preparations for the exam and then I can no longer remember everything."
"mit etw. spät dran sein"/"to be late for smth." is a set phrase, but the "dran" always refers to the "etw."/"smth." that comes before it. In this case, the "dran" is connected to the thing you're doing, i.e. the "Vorbereitung der Prüfung".