I was going to say what Nouw said. I'd give me the money because that way I win right? I'd try to shame him into not fighting.
If I needed to win though, luckily I'm really really twitchty and the first natural reaction to someone jumping at me would be to lash out hard, so I'd catch him and hopefully knock him to the ground.
From this point whatever I do, I win, because we've got identical abilities and I've just won the upperhand, as long as I go in slow (I'd probably tussle on the ground because I'm a pansy who can't throw a punch) there will be no way to win the engagement. It'll will be a slow war of attrition but it will end with alternate-me alternate-face-down with my alt-arm twisted behind my alt-back, and me sitting on my head. I'm probably really wishing that I'd sat down and thought of a cleverer way to engage myself...
...but I clearly did know I was going to fight myself, afterall I chose to ask myself for money rather than some random stranger (unless I expected me to just give the money on account of not enjoying robbing complete strangers, but I would try and help him out there), anyway since I knew I was going to fight, I'd either bring a weapon that I knew I wouldn't have since normal-me wasn't planning to fight, or knowing that a leap would trigger reactions and end in my defeat, he would go for a slow grapple first and use the tiny advantage to win...
... except he would be calculating and I would be afraid so I'm likely to have the greater adrenaline and win again. Knowing this alternate-me would have deliberately trying to hype himself up as much as possible, possibly seeking out some sort of drug to help out and he'd kick my arse...
...naturally I now know what alternate-me has planned to do the minute he asks for money and I know how he's prepared but although I have the space to outsmart him, I don't think I would have the ability in my situation. Unless I fought a war of attrition, fighting completely defensively, running away etc to delay the time until the drug wears off and he starts feeling a downer and then pounce...
...but now alternate-me knows that I'd prepare my tactics in a fight to beat him too and wouldn't take any drugs and instead use the advantage he gains when I try and run away to beat me...
... so I wouldn't actually run away but I would pretend to and then suddenly launch a counter-attack...
...which he would be expecting so he would knock me down onto the floor...
... but I'd be expecting the counter-counterattack, so instead I would dummy counter and then rugby tackle him when he incorrectly flinches...
... but I'd realise this is stupid and it's impossible to outthink myself. I'd way up how much I need money from myself, and if it's worth a 50% chance of failure, I'd flip a coin on whether I should attack myself first or dummy attack myself first, making outthinking impossible. Naturally normal-me would also do this and since none of us know if heads fall tails in the alternate universe or coins fall the same the result of our battle would come down to chance