Is there a way to refactor the c# code having multiple conditions?
You've got lots of tests for paymentAuto.PaymentType there, and many of the cases are the same. A switch statement will neaten that up, as it only names paymentAuto.PaymentType once, and it encourages you to put all of the cases nicely aligned together in code:
public static PaymentOptions GetPaymentOptions_Auto(TestConfigurationCDO testConfiguration, int siteId)
{
    var paymentOptionList = SitePaymentRepository.GetSitePaymentInfoBySiteId(
        testConfiguration,
        siteId);

    var lowestPriority = paymentOptionList.Min(x =&gt; x.Priority);
    var paymentAuto = paymentOptionList.First(x =&gt; x.Priority == lowestPriority);

    PaymentOptions paymentOptions;
    
    switch (paymentAuto.PaymentType)
    {
        case PaymentMethod.PayPalDirect:
        case PaymentMethod.Braintree:
        case PaymentMethod.BankTransfer:
        case PaymentMethod.AdyenDropIn:
            paymentOptions = new ClientCheckoutOptions()
            {
                paymentMethod = paymentAuto.PaymentType
            };
            break;

        case PaymentMethod.Klarna:
            paymentOptions = new KlarnaOptions();
            break;

        case PaymentMethod.PayPalExpress:
        default:
            paymentOptions = new PaypalOptions();
            break;
    }
}

The initial value you were assigning to paymentOptions wasn't used, so you can leave that variable uninitialised. The compiler will moan at you if you forget to initialise it somewhere in the switch statement.
That .FirstOrDefault() shouldn't fail to find a matching item. To make sure of this, I changed it to .First(), which throws an exception if it can't find a match. .Where(...).First() is the same as .First(...) (but a bit more expensive), so I changed that as well.
Annoyingly there's no .MinBy method in Linq (there are plenty of examples of implementations around, though). If you have this, you can write var paymentAuto = paymentOptionList.MinBy(x =&gt; x.Priority).