The 144 will have no bearing on a ps4.
It is only capable of 60.
If you are going to upgrade to next gen then yes it will. From what I have seen it will be set at 120 I believe.
If you are using a standard PS4 look for anything faster than 5 ms. Also HDR and 4K will help with graphics. The faster ms time generally is to distinguish gray to gray. The other graphics will make dark areas more vivid.
As far as lag hits no monitor or setup will fix it. It is client side lag, which has been discussed several times here. Basically you are not where you are on your screen on your opponents screen. It has gotten better. Perfect example you on your screen close a door and a magic kraber bullet kills you thru the closed door. Your enemy view would show the door still open hence you got shot through his open door. You die to a client side server issue, yours is faster his is slower. The system has to recognize which server is accurate.
I know way more information but we have all had to fight the rubber banding pathfinder.
If it was me I would but the monitor just to see this game in full clarity. It made a huge difference in my accuracy across the board. No streaking or blurring and no lag in snapping onto targets.