You can add your surname in your birth certificate in India by moving an application for the same before the authorities where your birth has been registered.
You can get the change of name changing your name ie. adding your surname after the said name as it exists. This will also solve your purpose in respect of your educational certificates in case they don't carry your surname in them.
If this is adopted then I advise you to purchase as many copies of the relevant news papers from the stands when they hit them as thereafter it will be impossible for you to get any such newspaper thereafter. Small expenditure and of extreme value in your life, this act of purchasing will prove.
You are advised to approach the authority where your birth has been registered and move an application for the correction of the same, stating therein that your surname is ………….. And want the same added in your birth certificate.
That this is a mere clerical error. Attach attested photostat copies of your credentials where your surname and your father's surname have been mentioned . Attach photostat copies of the above newspapers is no such certificate is available.
Draft and get attested an affidavit on the above lines and attach it with your application for correction of the same.
You are advised to obtain four or more copies of the corrected birth certificate as you may deem fit as the same will be of great use in your future.
Anyway get the above correction made and apply for the passport. Remember and keep in mind that it is always better to have a surname on your birth certificate and your passport also as the same foes help in life.