How to go from old Hong Kong ID card to smart ID card and/or passport?

Yes, you can apply for the HK passport and ID card at the embassy.

Documents needed for applying the passport:
•Form ID 843
•Three recent colour photographs with white background
•Fee payment form (Form ID 853) for HKSAR Passport

Documents needed for applying the ID card:
•Your birth certificate if you were born in Hong Kong.
•Proof of your previous residence in Hong Kong such as school documents, proof of employment, official receipts or bank statements if you have ordinarily resided in Hong Kong for a continuous period of not less than 7 years.
•Your birth certificate and parent’s permanent identity card if you claim the right of abode in Hong Kong by virtue of your parent’s status in Hong Kong.
•Two recent identical photographs
•The old ID card
•Form ROP144