How many AES-operations have to be watched for key-exploit by side-channel attacks? If I would like to exploit a used key by side-channel attack, one that measures electromagnetic fields, how many runs of the AES-128 do I need to spy on for a successful exploit?

I can't use any programs or software on that device. AES is software implemented. And I try (theoretically!) to spy in a distance of X meters. I am able to reduce the noise by surrounding emissions. But I can't have any type of access. So the attack model specifies using a sniffer. 

I do not need an exact number of traces. There are only certain values for trace/distance that I could think of. I just found the TEMPEST attack, which needs 400k traces in a distance of 0.3 meters and 2.3m in 1 meter.
As pointed out by poncho, it is not possible to precisely answer your question since too many parameters are involved:


About the AES implementation: is it an unsecured implementation? Does it use masking? If yes, at which order? And so on...
The physical properties of the device also matter, does it leak a lot? Are there some hardware countermeasures such as a shield, a jitter, etc...
What equipment is used to measure leakages? Depending on the probe, the oscilloscope and its setup, the number of AES executions to achieve a 100% success rate may vary greatly.


To put it in a nutshell, you have to make practical experiments to answer your question.