If Player 1 does not want to lose the game, one thing (and the only thing) he can do is to mortgage the (only) property that he owned. So although he will still lose $15 from this deal, at least he would not go bankrupt when Player 2 lands on his property.
Doing this move is a gamble as Player 2 might not land on his property at all (For obvious reasons, you can't do anything in the time interval between when the opponent rolls the dice and moves his token to the square). This includes mortgaging too. (Although I don't have a source for that, unfortunately, this should be quite obvious to understand -- otherwise there would be people instantly upgrading houses to hotels once someone rolls a 7.)
Though look at the chances of this happening -
The cheapest property would be either property in the brown set, right after GO, and thus if Player 2 was to land on it, he would have been able to pay the $2 or $4 rent after collecting $200 from GO. Thus, the ONLY chance of this repeated bankrupting happening as described is when Player 2 lands on Community Chest, then drawing the Go Back To Old Kent Road/Baltic card. If Player 2 rolled a 2 or 3 AFTER passing go, owing a $4 rent, Player 2 will definitely have either some cash or some unmortgaged properties, as he will collect $200 from passing GO. Thus, this case is impossible.
For any further hope of Player 1 winning the game, he has to survive up to one round of the monopoly board without paying a single dollar, and collect the $200 upon passing Go.
For the side question -
This situation occurred in a Monopoly Tournament
"Phil Orbanes, who wrote The MONOPOLY Companion and is the official Judge at the World Championships provided me with this answer when it came up in tournament play a few months ago:
If you cannot pay a debt to ONE player due to a card, you go bankrupt to that player and turn over all asset (std. procedure here).
If you cannot pay a debt to TWO or MORE players (after calculating any money you could get from the bank by mortgaging and selling houses) due to a card, you go bankrupt to them all. The issue then becomes: fairness. How do you divide your assets as evenly as possible? Procedure: after selling any houses for 1/2 price to the bank, the bank then rebuys your properties at face value if unmortaged, or for 1/2 value if mortgaged. You divide the resulting cash as evenly as possible, with the player(s) to your left collecting any odd dollar(s). The bank then auctions all properties it purchased to the highest bidders."
Source : Quora.com : Kevin Tostado, Director of "Under the Boardwalk: The MONOPOLY Story." MonopolyDocumentary.com AND a previous question that also discusses on this situation
Conclusion: The players may get a lot more than the $50 that they were originally owed.