No, you cannot apply for asylum at a Swedish mission abroad. In order to apply for asylum you need to have entered Sweden.
The best way is to submit an online application, for more information please refer to the Swedish Migration Agency's website: www.migrationsverket.se
In order for the Migration Agency to be able to assess whether you are entitled to move to Sweden, the child living in Sweden must be unmarried and under the age of 18 years, have arrived in Sweden without parents or any other guardian, or have been left by him or herself after arriving in Sweden and have received a residence permit in Sweden as a refugee or a person in need of protection.
For the Migration Agency to issue residence permits for family reunification you need to be able to prove your identity by showing your passport. Only in very special and rare cases can an Alien’s passport be issued by the Swedish Migration Agency.
The Embassy does not grant priority to individual cases, unless requested by the Swedish Migration Agency. When booking an appointment you will be given the first available time slot.
You may need a Schengen visa for entry. A Schengen visa is a permit required by persons wishing to enter and stay for maximum 90 days in the Schnegen area
If you have a complaint or suspect criminal offences or irregularities related to the Swedish Foreign Service’s activities, please report these to the Ministry for Foreign Affairs.