I made this trip last year in January in a Corolla GLi and I can provide some info as i too was unsure of starting time, but I think I made the right decision of leaving just before fajar. Left from DHA, Karachi at 5am, at around 6:30 I was crossing Hyderabad, took the Amri bridge detour to Moroo, that road was fantastic at dawn.
Moroo till Sukkur is a shit show, you will encounter alot of slow moving heavy traffic, bikes, chingchi rikshaw, etc. I eventually reached Sukkur at 1PM where I stopped for lunch at the PSO pump just before Rohri interchange, this was also the first time I topped up fuel since DHA Karachi.
After Rohri is smooth sailing on the motorway and gets boring real fast. I drove straight to Lahore with 1 tea stop on the motorway, after Multan. Got to Lahore at 9PM the same day, took me 14hrs. It was an amazing experience, more so because it was my first time.
If you have family with you, and they get car sick, a stop over at Multan would be advisable.