Okay! So! There is quite a few different things going on here. Alexander Nevsky was the Prince of Novgorod and then became the Grand Prince of Vladimir (The city, not someone's name).

Novgorod at the time of Alexander Nevsky (Aleksandr Yaroslavich "Nevsky") was a proto-Republic. It was ruled by a general assembly called the veche which decided a lot of matters. It was also protected and funded by an aristocratic class called the boyars. The boyars were the nobility in Kievan Rus, although similar to Knights in Western Europe, they tended to be slightly different. Novgorod would elect a Prince from a nearby other city to "rule" for them in the executive sense, controlling the combined boyar army and performed other executive power. He however could be deposed/exiled if he no longer was in favor. This is what happened to our dear friend Alexander Nevsky. He won a great battle (allegedly) against the Swedes in 1240, but they grew out of favor with the boyars. The Novgorod Chronicle, which is one of the Primary Sources for this period in Novgorod simply stated that after the battle with the Swedes:

"In the winter of the same year (1240) Knyaz (Prince) Alexander went out from Novgorod with his mother and his wife and all his court, to his father in Pereyaslavl, having quarelled with the men of Novgorod."

Subsequently he was called back to was deposed only to get called back again in 1241 to once again defend Novgorod against the Livonian Knights ( a branch of Teutonic Knights) at the Battle on the Ice. This cemented him in Russian folklore and spawned great tales and a great movie during the Stalinist period of the Soviet Union.

He would later be "elected" to be the Grand Prince of Vladimir which was the de jure ruler of a large portion of Kievan Rus at the time, under the Mongol Yoke. Which kind of leads to your other question, which is if Novgorod was an exception or the rule. At the time, Novgorod was the exception because of its unique history and also having not been directly invaded by the Mongols. Majority of the major Rus cities were taken by foreign invaders at this point, often leading to Princes being chosen by the rulers. This is how Alexander ended up being Grand Prince for he… metaphorically speaking, bent the knee. I am personally not sure if the veche and the election of Princes before the Yoke was common in other Russian cities. Also it is worth noting that in actuality the Romanov dynasty was elected by a boyar council after the Time of Troubles in the 17th century. However, I do think majority of the Princes other than Novgorod were typically dynastic and usually tried to draw lines to Ruirik, the original Grand Prince. These were affectionately known as the Ruirikids (it's the actual historical term. No joke.) I also know there is a lot of speculation about the veche and how it was, at the time, kind of revolutionary in European politics, being that it was a proto-democracy/republic in the age of feudalism. The veche however became weaker because of the Mongol Invasion which led to its deuse and the rise of Muscovy as the dominant power in the vacuum after the Mongol Empire slowly dissolved.

Edit: I forgot something! So, the veche was actually a bell in the commons area of the city/town. To form the assembly to vote on something they would ring the bell and the people would gather to vote. There was some rules on who could ring the bell and for what. But that was how it was run. Actually, I found the movie Alexander Nevsky on YouTube and at this point of the movie you can see the Soviet interpretation of the veche in Novgorod was. Here is the link!