First, you will have to contact an agency that can do Paper Cup Branding successfully or that has a successful track record in Paper Cup Branding and has previous experience with some good brands. Like you choose The Ginger Media for this because they have a successful track record and they have worked on 175+ projects in all of India. Apart from this, they have a good reputation in the field of branding. Then you will have to contact them and give them all the information related to your brand (such as brand name, brand logo, contact details, and other information which you wanted to include in an advertisement) which will be applied to the Paper Cup.
Although the paper cups are also arranged by the agency itself, if you want to get your paper cups made by someone else, then you should also talk to any paper cup manufacturer for that. But if you do this, you will have to give all information related to your paper cup such as materials other than paper used in Paper cups, availability of paper cup sleeves and handles, etc. because all these things can affect the printing.