The way I've interpreted this is that the spell itself, the casting of it, takes the duration of one round, your current action. The effect itself does not take place until your next turn as stated in the spell text proper. Once your next turn begins the spell is active, so to speak, and for the duration of that round you gain the advantage listed.
Furthermore if concentration is lost between casting and actual use, you'd resolve it as you would normally for checking on a break in concentration.
TRUE STRIKE
Divination cantrip
Casting Time: 1 action
Range: 30 feet
Components: S
Duration: Concentration, up to 1 round
You extend your hand and point a finger at a target in
range. Your magic grants you a brief insight into the
target's defenses. On your next turn, you gain advantage
on your first attack roll against the target, provided that
this spell hasn't ended.
So cast spell, wait for next turn, spell effect is active for that turn, profit.
Furthermore I'd treat an attack on that turn that a PC might be able to make as a bonus action as the "next turn" attack with advantage, aka the first attack roll against the target mentioned in the spell proper. But the text is pretty clear that it starts working on your next turn, so that would likely be up to the DM.
I'm pretty sure that inflict wounds is your best bet. The 5e spell list is small enough that it's possible to scan the whole thing, and there's nothing that deals more damage under the specific constraints you listed (pre-cast true strike, attack roll, not using any daily powers other than the ninth-level spell slot).
I've got some space left in this answer, so I'd like to address a related question. This is a question you haven't asked, but it's a question which some of the people reading this answer might be curious about: "I'm fighting something that has Legendary Resistance and will choose to succeed at any saving throws I offer. I have a buff spell pre-cast. What's the most damage I can deal, ideally without walking into melee range?"
It turns out that most of the good answers don't take advantage of the true strike. For instance, the old standby meteor swarm deals half of 40d6 on even a successful save, so that's 70 damage plus whatever bonuses you can get from modifiers.
Perhaps our character doesn't start out at level twenty; perhaps, rather than meteor swarm, we'd like a spell that works great using a ninth-level spell slot but also scales down gracefully. Rules As Written, it's worth looking at magic missile: there's an official, though unpopular, ruling that magic missile multiplies your bonus spell damage per missile. Combining this with Empowered Evocation (a wiz10 ability that adds your INT mod to your damage roll) and Hexblade's Curse (a hexblade warlock ability that adds your proficiency mod to your damage roll), can lead to very large numbers which are technically legal -- although, in practice, your DM is unlikely to permit this.
The true strike spell grants advantage on the first attack per round, but a twentieth-level caster might be willing to use a stronger buff, such as greater invisibility. Used against a creature that can't see through it, this spell grants advantage on all attacks in a given round. With this buff, scorching ray is better than inflict wounds: it fires ten rays maximum and deals 2d6 damage per ray, for an average of 70 damage, or 58.8 damage after applying the 84% hit chance.
Best Answer
There are several reasons why you might want to cast True Strike. The first is that depending on your class, you might be able to get a bonus action attack after casting it, giving you one attack with advantage and one without, a definite improvement on two attacks without.
The second is that you might use it to overcome disadvantage:
The third reason is that you might want advantage for something. For example, a Rogue might use True Strike to grant advantage to allow use of Sneak Attack.
Another reason is that you might know in advance that there was going to be combat, and cast True Strike for advantage on the first round. This would probably only happen when ambushing, but is still worth mentioning.
The final reason is that you might have an attack that you particularly wanted to hit. A Wizard using Plane Shift to send an enemy to the Abyss, for example, would want to be certain that the melee spell attack required would hit, otherwise they've wasted a high-level spell slot for nothing. Casting True Strike first would make it considerably less likely to miss.