In the end of modern warfare 1 price is rendered unconscious while you are conscious and somehow make your way back home. How did price end up in gulag while you - the player as soap mactavish were free to go considering that you were in the custody of the same people - the ultranationalist forces that ultimately saved you.
I is not completely certain that this happened this way since there are some holes filled by fans and rumors but this is the accepted story.
A Russian medic can be seen trying, seemingly in vain, to revive Price, but Soap and Price are taken to a Russian hospital.
Sometime after healing from their wounds sustained from the Zakhaev incident, both Price and MacTavish were reassigned to General Shepherd's newly formed-elite unit, Task Force 141. Acting as the field commander for the regiment Price led his new unit on countless covert operations with the key goal of capturing/killing Vladimir Makarov.
On October 8, 2013, Price led a joint Task Force 141/Delta Force operation codenamed Operation Kingfish, to raid an Ultranationalist base in Ukraine suspected to house Makarov. Price conducted the raid along with Soap, Simon 'Ghost' Riley and Delta Force operative Sandman.
Price was captured by Makarov's men and imprisoned in a Gulag east of Petrovpavlosk, Russia, and was designated "Prisoner 627". Price would remain imprisoned in the Gulag until 2016, where he was rescued by Task Force 141 when they attacked the site to recover him.
I don't know the whole answer, but I do know that he was in Russia when he was captured. At the start of Modern Warfare 2 when Shepherd is talking and the statue of Imran Zachahev is unveiled before the crowd, you will see a group of police wearing riot gear. Among these police is a man with the unmistakable mustache of Captain Price. I don't know what happened next, but he probably got captured shortly after that