As far as I know, you could use credit card when you want to buy daily goods at a mall in the city. You need to pay most of the things you buy in traditional markets and other smaller shops. If you buy an airplane ticket then you can't use credit card either. Most of the ticket agencies in this city only receive payment in cash. It means you need to change your money i.e. from US Dollars to Rupiahs. You can do this when you have arrived in Jakarta or Denpasar. The current exchange rate is 9,000 rupiahs for 1 US dollars. This value may change anytime.
Manokwari is a fast growing city after its status changed into the capital city of West Papua province. Several airline companies such as Batavia Air, Lion Air, Express Air and Wings Air have opened their offices in this city. I don't know whether they accept credit card payment or not.

Please contact the respective airline to find out about it. There must be information about flight schedules and ticket rates on the internet and also payment schemes. According to the website of Batavia Air, you can buy ticket for domestic flight from Jakarta to Manokwari using credit card. The online payment is powered by mastercard, and verified by visa. Online payment using klik BCA and Paypal is also available in Batavia air's website. You can also book your ticket through online reservation directly from their website. I believe that such service is also provided by other airline companies for domestic flight from major cities in Indonesia to Manokwari city.

A new airline company that provides flights to many cities and towns in Indonesia is express air. Its airplanes fly from Jakarta to Makassar, Sorong, Manokwari and Jayapura. There are other flights to cities and towns in Sulawesi, Maluku, and Nusa Tenggara islands. In its website, Express Air promotes Diving in Raja Ampat of with flight to Sorong as the main vocal point of tourist attraction in West Papua. As a matter of fact, the adjacent city, Manokwari also has many tourist attractions such as Caving, Bat Watching and Bird Watching in the Table Mountain, and Snorkeling, Scuba Diving and Shipwreck diving in the islands in the Dorey bay of Manokwari city. Most of the hotels and losmen in Manokwari do not accept credit card payment. Tourists have to pay in cash for the services that hotels and travel agencies give to them. by Charles Roring

Manokwari Travel Guide