Cut out two circles in both colors of felt, one long rectangle in the color that you chose, and one small rectangle in the light brown.

Begin by sewing the light brown circle to the long rectangle. I like to use the blanket stitch for this.

Cut off any excess fabric(save it for the next step!) and flip the cup inside-out. Sew the ends of the base together.

Use the felt that you cut off from the previous step to make the cup handle. You can do this any way you want, but here's how I make the handle;
Fold the felt in half and sew it in place. Sew one end closed, then add your preferred stuffing. Then, stitch the other end closed and sew both ends onto the cup(this way, the handle curves).

Stuff your cup with your favorite stuffing. Sew on the remaining circle.

Now we'll add on the teabag. Before you begin sewing, draw or write anything you want on the light brown rectangle. Add a cute little face, or sign your name on it. I like to write the kanji character for "tea" on it.
Once you've put what you want on the teabag, it's time to sew it on. Make one stitch on the top of the cup, then bring the thread down an inch or two. Sew all the way around the teabag.

Enjoy your new plushie!