Time to get off in a port can be extremely variable depending on many factors.
In many ports, the ship arrives at the dock and begins tying up before the published time, and pretty much right at the scheduled time everybody just walks off the ship.
In other ports there are formalities to be gone through, gangways to be placed, and so on. It is possible you might not get off the ship for half an hour or so after the scheduled time.
In tender ports it is a bit different. (A tender port means that the ship anchors off shore, and the ship's boats are used to tender, i.e. ferry, people ashore.) Generally, those on ships tours will report to a specified meeting place. For those not on ships tours, the usual procedure here is that at an announced time, perhaps an hour before tendering actually begins, they will hand out numbered tickets in a certain lounge.
Once the ship is anchored and the tenders have been lowered and all is ready, they will begin sending the tenders ashore. There is usually a ships tour that will go on the very first tender. Then they usually manage to interweave the ships excursion times so that passengers with tender tickets can go ashore in between ships excursions. It is NOT generally true that ALL ships excursions go before ANY independent passengers. It works pretty well and everybody gets a fair chance.
In a tender port, IF you get up to the lounge early and get the first tickets, it is possible to be ashore within half an hour or so of arrival. But that means really pressing things. Usually about an hour and a half after arrival, the tenders just run back and forth and no tickets are needed.
Also note that everybody waits til the last minute to come back to the ship, so there is usually a big line for the last few tenders.
If you are meeting an independent tour operator ashore, they are normally very familiar with the operations in that particular port and can suggest the time to start the tour that will allow you to get off the ship in time. If you just want to rush off to get ashore quicker -- well, honestly, my advice is to relax and take it easy.
PLEASE NOTE THAT THINGS CHANGE FROM PORT TO PORT, SHIP TO SHIP, AND SAILING TO SAILING. STUFF HAPPENS. All of the above might not apply to any particular port call.
Have a GREAT cruise!