And what do the AGS and Article 37 of the Constitution have to do with it? Or you didn't study well The federal law about AGS? Since when does a conscript have the right to choose a place for the ACS?

I study at a very good university and soon I will receive a diploma in construction. Therefore, I decided to choose an alternative to gain work experience during this time.
But I don’t want to be sent to work somewhere I don’t know. I want to get a job in a good one, albeit a government one, construction company(I want to go to Moscow, there’s also a girl there).

Eat special list organizations and positions approved by the Government of the Russian Federation, where it is possible passing the AGS. From this list they will select something for you. Of course, you can appeal any decision in court, but this is a long path with an unknown end.

Article 4. Place of alternative civil service for citizens

1. Citizens undergo an alternative civil service individually, as well as as part of groups or formations:
in organizations subordinate to federal authorities executive power;
in organizations subordinate bodies executive power of the subjects Russian Federation;
in organizations of the Armed Forces of the Russian Federation, other troops, military formations and authorities as civilian personnel.
Completion of alternative civil service in organizations subordinate to the authorities local government, is determined by federal law.
2. Citizens undergo alternative civil service, as a rule, outside the territories of the constituent entities of the Russian Federation in which they permanently reside.
If it is impossible to send citizens to perform alternative civil service outside the territories of the constituent entities of the Russian Federation in which they permanently reside, citizens, in accordance with the decision of the specially authorized federal body executive authorities may be sent to perform alternative civil service in organizations located in the territories of the constituent entities of the Russian Federation in which they permanently reside.
3. Citizens belonging to indigenous peoples are sent to perform alternative civil service in the organization traditional industries management and traditional crafts.
4. Lists of types of work, professions, positions in which citizens performing alternative civil service can be employed, as well as organizations where alternative civil service is provided, are determined in the order established by the Government Russian Federation.
5. When determining the type of work, profession, position in which a citizen sent to alternative civil service can be employed, and the place of alternative civil service, education, specialty, qualifications, and experience are taken into account previous work, health status, Family status citizen, as well as the need of organizations for labor resources.
6. Labor activity citizens performing alternative civil service should not interfere with the employment of other persons, and also serve as a basis for the transfer to another place of work of persons performing work on employment contract, or for their dismissal.