The Time Eater is basically a hard counter to card draw and rotation style decks, unlike any other boss or enemy in the game. If you’re relying heavily on this strategy in Act 3, the outlook is bleak.
Trying to play a consistent number of cards per turn may be useful, so you don't end up with a 1 play turn because of the Time Eater's Time Warp ability. For example, it's preferable to play 4 cards, then 5, then 3, rather than 7 cards, then 4, then 1.
The heal action can only happen once, and will always happen the turn after falling to half HP. Since it erases all debuffs and heals to half HP, there is no reason to take offensive action once you pass half HP unless you can burst the Time Eater down in one turn. It is better to play Powers and buff yourself for the remaining battle, when possible. Due to the fact that the heal can be reliably activated, it may be worth ending the turn with the timer close to 12 if it lets you get time eater below half, as you can safely play only one or two cards next turn while he heals, resetting the timer.
Try to avoid using low-cost, spammable cards when you can. This is important because those cards will quickly increment the Time Warp ability and the Time Eater will gain too much strength to be dealt with. For example, if you play Acrobatics, draw 1 Strike+ and 1 Defend+, and play them both, it will cost 3 Time Warp charges. A single Dash+ would give roughly the same block and damage for the same amount of energy, but only 1 Time Warp charge.
The Watcher can nullify Time Warp by playing Vault as the twelfth card. The result is the same as if you had normally played a Vault card and the Time Warp counter is reset.
It is possible that Time Eater is related to the Watcher in some way. Their attires are different shades of purple and adorned with a similar motif. It also says "Never... Liked... You..." on its first turn when facing the Watcher, instead of the usual "Ah... company."
The Watcher's Sensory Stone text implies that her peer went missing and stopped reporting back to their sect about the Spire. Time Eater might be that missing peer of hers.
The Time Eater greatly resembles the Tonberry from the Final Fantasy series.