Is it possible to use a ASIC miner for that attack?
An ECDSA breaking ASIC? Sure. A Bitcoin mining ASIC? No. ASICs do one thing and one thing only.
How do I get from X bit code to Y bit security? (I assume the way to go is: find the best available algorithm for the problem and use the big O approximation) Not sure if this applies for this example since the way I understand OP we dont have the public key.
That is my mistake. My numbers assumed the PubKey was known. I misread the OP. If the PubKey is unknown then the best attack is 2^160 (searching for a preimage of the PubKeyHash).
How do I get from Y bit security to Z calculations. IIRC big O does this as well, but ...
You use Big O or for ECDSA you just remember that bit strength is 1/2 key length. Also remember Big O just gets us to the magnitude. There is no computer which can perform Pollard rho in a single operation. It might require 10 operations or 10,000 but these are linearly constraints. For cryptography we want to ensure the magnitude alone puts an attack beyond what is feasible.
I dont really know what my but is here, besides that I am a little afraid because I just realized how powerful the sabre cluster actually is.
Against 64 bit security? Nobody should be using anything with only 64 bit security. Note this is supported by emperical evidence as well. A 112 bit ECC key (56 bit security) is the largest known ECC key to be brute forced. It was completed in 3.5 months of runtime using 200 Playstation 3s.
http://lacal.epfl.ch/112bit_prime