The more dire the situation became, the more calm Lex had to be. Realizing that he was, very literally, running out of time, Lex had to reassess how to go about this challenge, or he would soon die.
For him, the threat of death this time was no greater than any other time he had faced death. For example, being crushed to death by the aura of a dragon might feel different than being erased from time, but both objectively ended up with the same result.
That is why, although the danger was from a much more powerful source than Lex usually faced, the pressure he faced was actually the same. Not only had Lex learned to tolerate that pressure, he somewhat enjoyed it as well.
Now that he was, more or less, on the path to becoming the strongest being in existence, not that he had decided to pursue that goal yet, he would constantly be faced with death, so there was no reason to be so apprehensive about it.
