Some plants are like that. Leggy in the beginning and get stronger later with more light. Too much too soon ain't never good.
I would just gently prop up the baby. First I would stick in a plastic stick on side of the cup. Avoid the center as it may damage root. Plastic stick like cocktail stirrer, cut up cloth hanger wire, plastic chop stick. Don't use wood, it can draw mold.
And don't tie the plant to the stick, just let it lean on the stick. If you tie the plant, you make break and kill the poor thing.
And you can fill the cup to the rim. This will shorten the stem a little. Don't worry about overflowing water. On the first few weeks, you seedling does not need a lot of water. Water so gently with very little amount. Like a couple of thimble full each day. That is all the baby need. Too much water will kill seedling also.