Very well. I psychokinetically lift the plane onto a giant spring and note the deformation of the spring, this being the closest to a scale allowed in the exercise. Or, I dunk it in an enormous tank of water and note the displacement. Or, I grant it the power of self-knowledge and speech, and ask it what it weighs. (And if it's reluctant, I bribe it with some AVGAS.) I push it with an exact number of newtons and note the distance that it is pushed. Hypothetically, with some physics, any of these can give me the airplane's weight, or at least mass. (Weight=mass*pull of gravity.)
But granted omnipotence, I'm not stopping there. I'm not giving the powers back. Any attempt to take them back will result in me going back in time and beating up one of your ancestors until you cease such attempts. (Let's see your ancestor reproduce with a cracked rib and sore gonads.) And I'm not taking the job, either, because I can now create whatever I want ex-nihilo and no longer need money.
Although for fun I may attempt money making problems anyway. I buy a block of aluminum, go back to the 18th century, and sell it for lots of money. Aluminum was expensive back then because refining it from bauxite was very difficult. I then buy a metric insane amount of lobster, which was cheap at the time, and take it to a modern-day fish market. Any leftover 18th century currency can fetch a considerable price from a numanist, I anticipate a profit of at least $1,000,000 per cycle.
I produce for myself a luxurious house with considerable automation, and use my time-travel funding to pay people to do stuff that I think should happen, including a massive expansion of the international space station, and a floating Venus colony. I also pay for a number of software projects. With the last of my money, I operate a shadowy conspiracy for good fortune, paying people to make the world a better place, and when I run out, I manifest a bar of gold which I sell to keep up the funding.
Of course, with any insane enough premise, you can get any insane result, like Bertrand Russel's famous one:
If zero equals one, I am the pope.
Let us add 1 to each side of the equation, producing the equation 2=1. The pope and I are two separate individuals. But since 2=1, we are actually the same person. Therefore, I am the pope.
I guess omnipotence is just too insane a premise.