You always want to downplay the significance of the gift..... the standard Japanese to Japanese phrase for this is......
"Kore wa tsumarinai mono desu ga, demo dozo......." and hand it to them.
It means this is really a little nothing, but here you go.
Not too many westerners know or use that one in that setting. It usually surprises a Japanese person that you know to say that.
Don't expect someone to open a present you have given in your presence. Although if they are trying to be "western" sometimes they will, because they know that you likely will "expect" it. Usually it is opened later privately. Because you are a westerner, they likely will expect YOU to open something though.
As to the timing....... just use your "intuition" about the right timing.
best,
..................john