All state services related to passports (and much more) can be done on the Web site of the State Migration Service Of Ukraine, https://dmsu.gov.ua/
Basically, requests to issue/renew both the ID and the International Passport are the first two services they offer (highlighted by mine).
Caveat: Make sure the language selector in the top right corner is set to Ukrainian. If it opens in English (like I did while trying to make an English-language screenshot), the set of services will be targeted foreign citizens.
The sequence is:
- request a time slot in electronic queue ¹;
- decide if you want it biometric or not, "fast" (1 week) or normal service (3 weeks) as the fee would be different;
- go to bank and pay the fee ²;
- visit the DMSU unit, provide ³ with your ID and receipt;
- make sure your name is Latin-transcribed the way you like it. I have the "old" way of transliteration, and I have many visas, accounts, and international contracts signed with that name, so I had to ask for the spelling like it used to be.
- wait for 3 weeks;
- pick up your passport.
¹) The online queue is a new feature so it is available in a limited number of DMSU units, mostly in Kyiv and several large cities. If you reside elsewhere, the site contains a list of DMSU units you have to visit personally.
²) Don't use Web-banking for payment because you would need a stamped receipt. Just go to any bank, say you're paying for passport, and they will know where to send the payment. Pay with cash and keep the receipt.
³) The complete list of documents, state fees, and other conditions is specified on this page (in Ukrainian). Basically, an adult with no kids to be written in passport would need nothing more than local ID, an electronic or written request, and a receipt for payment the state fee.