For anyone who's played Teamfight Tactics a long time, this exact scenario has happened to them: you struggled early on but pulled together a killer team with only ten health left. Then you go on a seven-win streak and think you have a prayer of breaking into the top four. However, the one team that counters you appears on the screen and you couldn't position yourself in a way that might help you survive. Your board is decimated and, instead of a sweet LP boost, you end up sixth and miserable. If only you could've known that counter team was coming, right? Or, was there a way to know that team was about to destroy you?
The short answer is no, you can't tell who your next opponent will be. The randomness of who you might fight is a core game mechanic of TFT. However, that randomness is fixed, so you tend to fight people in a cycle. If it feels like you haven't faced a certain team in a really long time, they very well might be the next board you appear on. If you're particularly intimidated by them, it might be a good idea to position against them.
Other than some feelings and guesses, though, there is one way to definitively know who you'll be playing against. Future Sight.
Future Sight is an augment that lets you see who you fight next and gives you a Zephyr (or Radiant Zephyr, if you get Future Sight II). While this isn't as exciting of an augment early game, it can be absolutely game-changing in the mid to late game, especially if you are facing off strong, position-ruining teams like Assassins. Being able to position properly against certain builds can be the difference between sixth place and ending up in the top four (if not higher). Therefore, if you find a lot of value in preparing your board for each enemy, Future Sight might become your favorite augment in the game.
If you can't get Future Sight, there are a handful of other ways to try to counteract the randomness of who you fight. While you might not know exactly who you're up against, peeking at other boards is still invaluable. It means you'll be able to tell where the majority of people place their units, what they're building, and who their carry units are. Then, you can put together a positioning plan that gives you your best shot against the general trend of your lobby.
For example, if a lot of people are building Mages, your tanky team might do a better job closer up to the heavy hitters and try to break their formation. Meanwhile, if half the players are buying up Cavaliers or Assassins, it might be smarter to bunch up your positioning and protect your back-line as much as possible. It all depends on the type of players and boards you run into.