If I hold Reckoner’s Arena with Yone, Blademaster, will I place his triggered ability on the chain when the hold effect of Reckoner’s Arena resolves?

No, you will not. The main thing you need to know is that activating a conquer effect such as the Reckoner’s Arena effect is the equivalent of checking the condition of all of the specified conquer effects, as if a conquer had occurred at Reckoner’s Arena. Any triggered abilities of units at Reckoner’s Arena that trigger on conquering will see a conquer event occurring, but they will still need to check their other conditions in order to go on the chain.

In this case, the rest of Yone’s conquer effect condition is “...that was previously uncontrolled,” and since the battlefield wasn’t previously uncontrolled, Yone’s ability will not trigger.

Note that no actual conquer occurs—we’re simply treating the game as if a conquer had occurred at that battlefield for the purposes of the specified abilities. In Reckoner’s Arena’s case it specifies units, so only units with conquer effects at Reckoner’s Arena will see this conquer event and potentially trigger.

(We recognize the rules for “activate” in the way discussed above are underdefined. Expect further revisions in the next rules update.)