Is it improper to give a gift to a coworker who was recently promoted to be above me?

Check with your HR department.

Many companies actually have specific programs or mechanisms for "co-worker appreciation", which could be certificates, gift cards, nomination for the "president's award" or whatever. Even if they are not material in nature, they do help the other employee as it feeds into their performance reviews.

If your company has no such thing, ask them for what they feel is "appropriate" to do. Then follow their advice. If HR says a gift card or free lunch is fine, than feel free to go ahead.

If you do, let the other person know that you have cleared this directly with HR. This way they know that it's perfectly "clean", that they are no hidden motives or agendas and it makes it less awkward.

Send him an image with a blinking "congratulations" over an image of a package.