If I remember correctly it only looks at the total number of victories that you have. This was many years ago when I played the game but I distinctly remember only have to do like 1 additional to get to 5 once I got this quest.

The game checks the number of Knight difficulty arena victories you have when you try to turn in the quest (look at the Awards page to see how many victories you have). Page, Squire, and Lord do NOT count. It doesn't matter when you get them, as long as you have 5 when you try to turn in the quest.

You can't complete a quest before you receive the quest, but you can (sometimes) fulfill the quest criteria before getting the quest. In the case of the Champion promotion, you can go ahead and collect the 5 Knight victories, then you can acquire the quest (once you are eligible for it) and immediately turn in the quest for your promotion. You may have the victories, but you won't become a Champion until you actually click to turn in the quest.

Some quests can't be "pre-beaten" like this. For example, you can't acquire the Perfect Bow for the Master Archer/Sniper quest before getting the quest simply because the bow doesn't spawn until you've been given the quest. I believe the Crusader promotion is similar, in that Wromthrax won't be around to be slain until you've received the relevant quest. You can't use the broken belt for the Warrior Mage quest until you get the quest (and the belt).

You CAN go kill everything in the Haunted Mansion before receiving the quest to do so, and then turn in the quest immediately after receiving it.

Some tips to help with getting those Knight arena victories:
-Stop leveling up until you've got all 5 victories. As noted by the other user you mentioned, you'll be facing stronger (and more) monsters with a higher level party.

-Once you are in the Arena, do a few things before selecting a difficulty: 1) Cast all of your available buff spells, 2) Wait for everyone to be ready to act, and 3) Go into Turn Based mode. That way, everyone will be ready for combat and you'll have a chance to look at your opposition before they start coming after you. If you have invisibility, then use it; it will let you move to a corner so you don't aggro all the monsters at once.

-Make use of the 'terrain', such as it is. Use the ramp to block some monsters from reaching you (get them stuck on the other side), though be aware that flyers will eventually work their way up and over. Use the rising and lower platforms to block ranged attacks from monsters and to slow down some of the melee monsters (who may have to walk around it).

-Hire a Horseman hireling in Harmondale. The Horseman reduces travel time, allowing you to go to the Arena, fight, return, and you only have to rest/wait one day to immediately go again. With a Horseman, you'll return on Saturday on the same week, instead of Thursday of the following week.

I do this quest early to avoid the Manticore Bug.
But I never used the Horseman to speed it up. I will try it next time.
Thanks

Manticore Bug?

Sometimes an invisible Manticore fires arrows at you in the Arena. Killing it makes the Manticore appear and the game crashes. It was probably a branch of the game that never made it to the final version but the monster was never removed from the Arena roster. It only happens in the tougher arena fights. But after dealing with a tough bunch of enemies to have the game crash is a little too frustrating for me. I stick to the lower levels to complete the quest.

Ok, so I finally came so far as to actually diong the quest and acan confirm it works. The most efficient way seems to hire an "Explorer" NPC which cuts down travel time by one day, making the journey TO the Arena take three days. However, the journey back takes four days making you both leave and arrive on a Sunday.

Hey, nice! I'll have to remember that one. Half the price of the (already cheap) Horseman, too.