Automatically claiming draw when low on time Suppose X and Y are in an endgame:

X has a rook and 400 seconds.
Y has a rook and 40 seconds.

Y obviously cannot claim draw because X can theoretically win, even
though a minimally careful play would result in a draw.
X will play to make Y's time run out, but has to avoid the 50 move draw and
the threefold repetition draw.
Now, since Y is hard pressed on time, he may not notice that a position has
been repeated three times. More inexperient players may even overlook that
the 50 moves have been made. In this scenario, will Y actually have missed the
chance to get the draw? Or can Y tell the arbiter that he wants to have the
draw if any of those events occur, even if Y misses it?
Lichess has an option to claim draw on threefold repetition when time remaining &lt; 30 seconds, thus I'm wondering if there is something of the sort for on the board games.
I'm wondering if there is something of the sort for on the board games

The closest thing to this according to the latest FIDE Laws of Chess is article 9.6:

9.6 If one or both of the following occur(s) then the game is drawn:
9.6.1 the same position has appeared, as in 9.2.2 at least five times.
9.6.2 any series of at least 75 moves have been made by each player without the movement of any pawn and without any capture. If the last move resulted in checkmate, that shall take precedence.

However this process is not automatic. For either of these scenarios to play out then either the arbiter has to be watching and counting or the moves have to be recorded by at least one of the players and then brought to the arbiter's attention.
In general a game is drawn after 3 repetitions or 50 moves without a capture or pawn move only if one of the players makes a valid claim backed up by a recording of the moves.
See:

9.2.1 The game is drawn, upon a correct claim by a player having the move, when the same position for at least the third time (not necessarily by a repetition of moves):
9.2.1.1    is about to appear, if he first writes his move, which cannot be changed, on his scoresheet and declares to the arbiter his intention to make this move, or
9.2.1.2   has just appeared, and the player claiming the draw has the move.

and

9.3 The game is drawn, upon a correct claim by a player having the move, if:
9.3.1 he writes his move, which cannot be changed, on his scoresheet and declares to the arbiter his intention to make this move which will result in the last 50 moves by each player having been made without the movement of any pawn and without any capture, or
9.3.2 the last 50 moves by each player have been completed without the movement of any pawn and without any capture.

but note:

9.4 If the player touches a piece as in Article 4.3, he loses the right to claim a draw under Article 9.2 or 9.3 on that move.

So, you can't retrospectively claim a draw under 3 fold repetition or the 50 move rule.
Note that if there are no increments then in special circumstances you may be able to claim a draw when low on time in endgames like KR vs KR.
Here is what the rules say:

III.2.1 The Guidelines below concerning the final period of the game including Quickplay Finishes, shall only be used at an event if their use has been announced beforehand.
III.2.2 These Guidelines shall apply only to standard chess and rapid chess games without increment and not to blitz games.

Important to note that these rules never apply to blitz games, only standard and rapid. Also important to note that they only apply if announced beforehand. This will rarely happen.
Then:

III.4 If the player having the move has less than two minutes left on his clock, he may request that an increment extra five seconds be introduced for both players. This constitutes the offer of a draw. If the offer refused, and the arbiter agrees to the request, the clocks shall then be set with the extra time; the opponent shall be awarded two extra minutes and the game shall continue.
III.5 If Article III.4 does not apply and the player having the move has less than two minutes left on his clock, he may claim a draw before his flag falls. He shall summon the arbiter and may stop the chessclock (see Article 6.12.2). He may claim on the basis that his opponent cannot win by normal means, and/or that his opponent has been making no effort to win by normal means: