A check you get for 1099 work is technically not a "paycheck". The word "paycheck" means a check for W-2 work. This has nothing to do with taxes, but it is important to be aware of it, in case it comes up later.
For federal self-employment tax, estimate $60 per week.
For federal income tax, if your total income for the year, from everything, not just this, is $19,000-$20,000, then estimate $1200-$1300 per year, which is $25 per week. If you also have other income, then the tax on the income that we are discussing will be higher. (The tax rate on all the income depends on the total amount of income.)
Therefore, the total to save per week for taxes is between $80 and $90 per week.
To convert this to a percentage, divide by the amount of income ($400), which gives you 20-22.5%.
This makes since self-employment tax is 15.3% of 92.35% (which comes out to 14%) and income tax is $0 on approximately $9400 and 10-15% on the rest (which averages out to 6-6.5% for income tax), so the total should be 20-22.5%.
You are not allowed to hold onto the money until the end of the year. You must pre-pay the estimated tax four times per year. See Form 1040-ES and the instructions for Form 1040-ES to learn how to pay the estimated tax and how to calculate the amount to pay.