If you look closely at Certificate of Discharge, you will see that your man's rank for this voyage is
Mate.
This would mean that he was effectively second in command.
The reason for him signing in place of the Captain can be explained quite simply. The Captain may have died or been taken ill. He may have been on other business or drunk and incapacitated!
The 1869 logbooks and crew agreements for ARROW Official number 20837 in particular the one covering 8 January, which is the one you want, are stored in Newfoundland.
You can get hold of a copy from.
https://www.mun.ca/mha/holdings/viewcombinedcrews.php?Official_No=20837These documents should reveal the true reason for him signing his own Certificate of Discharge.
The Shipping Master was probably the Harbour Master at Bristol but may have been the Captain of ARROW you can verify from the L/B and C/A.