Right now (July 2015), the current record for RSA key breaking is for a 768-bit modulus, and it took two years of computations with a lot of machines, and, importantly, with some very fine brains too. A notable point is that the best known RSA-breaking method (that was used for RSA-768) is integer factorization with the General Number Field Sieve; GNFS does not benefit from any kind of "precomputation". This means that the people who factored the 768-bit RSA key, if faced with another 768-bit RSA key to break, would again have to spend two years on the job.
As for 1024-bit RSA keys:
- they are currently unbroken;
- they could be broken with existing technology, but it would require throwing at the problem a non-negligible number of millions of dollars to build a dedicated machine that would be good only at breaking 1024-bit RSA;
- ... and even that machine, if it is ever built, would still need a lot of time (months, possibly years) to break a RSA key.
In your case, if signatures only need to resist for a few minutes, then you could generate a new RSA key every day, and this will defeat GNFS-based attackers as long as breaking a 1024-bit RSA key takes more than a day, a property which is quite likely to hold for the next few decades. A good point (for you) is that since your signatures become obsolete quickly, you "just" have to keep track of scientific and technological advances: the day some people succeed at breaking a 1024-bit RSA key, you can switch to 2048-bit keys; you do not have to do it in advance.