Asian Porcelain Container with Small Cup Red Mark on Bottom

I have a Japanese porcelain container and cup with cover and I'm trying figure out what it actually is - i've been told the characters of the writing are "old japanese" but not sure.

It is 2 1/2" tall & 2 1/2" diameter without the domed lid.  The domed lid adds ~2 inches to the height. When you take the lid off. There is a small cup inside ~2" tall 2 1/4" diameter that tapers to a base of less than inch. The little cup sits inside the container. It looks like something you might put warm water in to keep whatever is in the small cup warm? The little cup is a thin white porcelain. The larger container is also porcelain with a bird sitting on a branch. The top of the lid looks to have something similar to a carrot on top to use to lift the top off. There is lettering on the side of the container and on the top. The stamp no the bottom is a red stamp