How can I fetch the options or details of the shipping method configured in the admin side?
You can get that just like you get any config value:
Mage::getStoreConfig('PATH GOES HERE');
In you case it will be
Replace '[shipping_method_code]' and '[field_name]' with the appropriate values.
as @nevvermind said in the comments below.
If you have the shiping method model instance you can use:
take a look at how