Once a creature stands up they no longer suffer any effects from being knocked prone
Your reading is correct. A creature can get up from prone on their turn by spending half their movement, and once they're standing up again they don't suffer any further consequences from having been knocked prone. As you've observed, with an unhelpful initiative order, an enemy can recover from being knocked prone before an ally can take advantage of the effect (a common problem for any effect that an enemy can recover from or save against again on their turn).
Ready actions to get around the initiative order
Taking advantage of the Ready action is the best way, within the game's normal rules, to get around this problem. If the barbarian knows that the monk is going to try and knock down their foe, he could Ready an action to attack after the orc is knocked prone, and so the orc doesn't get a chance to get back up before the barbarian is upon them.
Readying does require that the character specify some sort of perceptible trigger - so you can't use triggers like "after the monk's turn" or "before the barbarian goes" - but you could trigger on things like "after the monk attacks", or even use verbal indicators like "attack when the monk shouts 'go'".
It's important to note that though it can be situationally useful the Ready action is mechanically expensive - it requires the use of your reaction plus it denies you the use of any ability which you can only use on your turn (like Extra Attack or Martial Arts), so it's often not worth it (especially for higher level characters). If your barbarian can normally make two attacks in an action, it's generally superior for them to get to take two normal attacks against an enemy than give up their reaction and make one attack with advantage. The only case I can think of where Readying actions in this way would be consistently useful is for a Rogue waiting for allies to move into position to grant them sneak attack - since a rogue will usually only be able to make one attack a round anyway and their bonus action abilities don't depend on doing something with their action.
In 3rd edition, getting up from prone was an action that provoked attacks of opportunity, so a target knocked down would be forced to risk taking attacks as it tried to get back up. If it bothers you that a quirk of the initiative can mean that the monk could knock down a target but they immediately stand back up before any ally can take advantage, you could rule that getting up from prone provokes opportunity attacks just like moving out of reach does; this obviously makes any ability that can knock people prone more powerful, though.