How come it's impossible for him to have sage mode now? Without even training he got into Kyuubi mode, seeing Naruto use Sage mode might have helped him to figure it out how to use it. It seems like Minato is quick learner. I'm not Minato's fan, I like him but that's it, I wasn't surprised to see him use sage mode, after all he was a hokage learning tricks shouldn't be hard for him
I'm sure people have better explanations, but I think like that. To begin with Minato had contract with frogs, I'm sure he had information about sage mode. And in Edo-Tensei he can't turn into frog I believe so one more reason for him to be able to use it? 