Here is a situation:

Initiative 15 - Monk attacks orc causing the orc receive the "prone" condition.

Initiative 13 - Orc stands up, ending the prone condition, and attacks the monk without disadvantage.

If a barbarian attacks the orc at Initiative 14, he has advantage. If the barbarian attacks at Initiative 12, he does not have advantage.

Reading Player's Handbook on prone (p. 190, 292) I don't see any effects of the prone condition lasting after the target stands.

Is this a correct interpretation of the prone condition and how standing cancels it?

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.

House ruling

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.

You're playing the rule correctly.

Unfortunately, there's not a lot you can do to fix this situation. I'm reading between the lines a bit and assuming that a monk knocking people down is doing so using the Open Hand Technique as part of a Flurry of Blows.

While you could ready an Attack action against the Orc, attacking outside your turn means you don't have a bonus action to spend on a Flurry of Blows, so you won't be able to trigger the Open Hand knockdown. You could give up one of your attack rolls to attempt a Shove (Player's Handbook p.195), but that Shove would then be based on your Athletics check rather than your Dexterity or Wisdom scores, and anyway you'd be giving up damage to do so, so that probably isn't what you intended either.

In this particular case, you probably can't use your knockdown for much of anything useful. It's an unfortunate consequence of the initiative system and 5th edition's insistence that you don't change initiative order after it's been rolled. If it's a big problem, you might ask the DM to consider alternative initiative methods, such as Side Initiative (Dungeon Master's Guide p.270), or your DM might be willing to allow a Delay that permanently moves you down in the order.

Yes. Prone condition ends when the character stands up.

If if the monk wants the prone condition to last through everyone's turn, he can use ready action to attack after the orc has ended his turn.