I would love to see both mini-games implemented eventually, but every now and then I see new players notice/complain about the lack of PvP on this server. With that in mind I would vote for Last Man Standing to be added first. LMS will also have much more replayability than Castle Wars. Castle Wars IMO is a mini-game that booms for only a few weeks or so, maybe a month, but afterwards becomes only played by new players who want the cosmetics, or possibly old players trying to get achievements (assuming CW-related achievements would be added). Obviously there are multiple ways of combating this problem, such as having CW points being 1.5x on the weekends much like Pest Control, adding new achievements like "Win X Castle Wars games on the Saradomin/Guthix Team", "Win X Castle Wars games in a row" etc.
However LMS is a gamemode that I think would flourish forever. Just the entire concept of it would bring a whole new flavour to this server that most others probably don't have. The reason LMS has more replayability than CW IMO is because of the fact that in CW players are going in with their own weapons and armour. Every match will most likely be the same gear, on the same side, fighting essentially the same people. Sure there is teamwork and coordination to be had and Clans can compete, but in LMS, you start with basically nothing, there are no sides (unless you make truces but ofc there can only be one winner) and you must loot weapons and gear scattered all around the game zone making it a new experience every time. Also since in LMS every player's combat stats are boosted to 99 for the duration of the game, that ensures that new players can join instantly after creating their character.
TL;DR
Last Man Standing > Castle Wars (In my opinion)