but I lost a couple fares for scraping the letterbox upon docking. That happens sometimes in the big ships.
Big ships don't have shields?
I took several missions with my Orca where they didn't want hull damage and even though I scraped station and even exited station in bumber-car style they never lost happiness factor or refused to pay.
Anyways... going to the core and how long you need depends on a) distance, b) your ships jump range and c) rough guesttimate of how much time you need per jump.
E.g. if going to the center you have to travel roughly 25.000Ly, and another 25.000Ly to get home so totally 50.000Ly.
If you ship can jump 20Ly per jump you will need 50.000 / 20 = 2500 jumps. If you spend 2 minutes for each jump (enter system, align with next system to jump to and spool up hyperdrive to warp out, as well as a small amount of time added to cover refuelling which will be needed) then it will take you
2500 jumps * 2 minutes = 5000 minutes / 60 minutes = 83,4 hours. If you play for 2 hours each day it will take you 83,4 / 2 = 41,6 days to complete the full trip. (Note that this is given from an average of 2 minutes per jump)
If you spend more time, e.g. 4 minutes per jump due to refuelling or other things then your total time is doubled and it will take you 83,2 days to complete the full trip.
We can also reverse calculate this. 2 weeks 6 days = 20 days available. To cover 50.000Ly in 20 days you have to do 50.000 / 20 = 2500Ly per day. With a jump range of 20Ly you have to do 2500 / 20 = 125 jumps each day to complete your objective. If you manage 1 jump per minute that will only require 2,08 hours per day. 1 jump per 2 minutes and you need 4,16 hours per day.