There is not a way to apply for citizenship through your child. There may be a way to apply for residence or green card status if you do not already have that, and your child is at least 21 years old.
You cannot directly obtain your citizenship through your child. Your child would have to petition permenant residency for you after he/she reaches 21 (by filing the I-130 Form).
Once you obtained your permenant residency (i.e. green card status) through adjustment of status (by filing the I-485 Form), you will have to meet the naturalization requirement to obtain US citizenship (by filing the N-400 Form). You can apply for naturalization 5 years after receiving your green card approval. Out of the 5 years (60 months), you should live in the US for at least 2.5 years (30 months). You should not have long and extended stays overseas (i.e. over 6 month period) during these five years. Prior to filing your N-400 Form, you should consult with an immigration attorney if you have violated any law during this period.
As previously stated, citizenship is not something your child can petition for you directly. As to filing for permanent residence however, there are two stages: your (21 years of age or older) child's petition, and your application for permanent residence. If you are in the United States, your ability to obtain the green card will depend on many factors, including whether you entered the US legally, if you had multiple entries, if you have criminal issues, among many other factors. Generally, however, if you don't have any specific bars to adjustment, you can get the green card through your child.
None of the contributors above addressed the second part of your question - your husband is a US citizen. If you do not already have a green card through your US citizen husband, you are immediately eligible to apply for one, and it will be a full 10-year green card. If you came to the US as an immigrant (lawful permanent resident/green card holder) when you moved back here from Japan, then because you are married to a US citizen, you will be eligible to apply for citizenship within three years after your entry as a resident.