When the US put a man on the moon it was a big thing in the US and for the west in general. But how was it in Russia. Did people feel a sense of pride outside Russia in the other USSR states or did they care much at all?
As we in west thought it was "cool" and what not when the US landed a man on the moon, but it didn't give raise to any special national pride of being a westerner etc.
Gagarin's flight was in 1961. This is some 16 years after the conclusion of WW2, and 44 years after the revolution. By this time the USSR was very much a single, centrally run nation, not a collection of states.
Having said that, there area few interesting things about the mission to note. It was carried out in secret and no announcement was made prior to the mission. Once the mission was a success, a press release was issued. There was nothing planned as far as a celebration, and the subsequent events were pushed by Khrushchev himself. According to his son (http://news.bbc.co.uk/hi/russian/russia/newsid_7015000/7015537.stm):
«Он начал с того, что позвонил министру обороны маршалу Малиновскому и сказал: „Он у вас старший лейтенант. Надо его срочно повысить в звании“. Малиновский сказал, довольно неохотно, что даст Гагарину звание капитана. На что Никита Сергеевич рассердился: „Какого капитана? Вы ему хоть майора дайте“. Малиновский долго не соглашался, но Хрущёв настоял на своём, и в этот же день Гагарин стал майором».
(My quick translation) He started by calling the Minister of Defense, Marshal Malinovskiy, saying that "He's a Senior Lieutenant, he must be instantly promoted." Malinovskiy said fairly reluctantly that he'll give Gagarin the rank of Captain. To this Nikita Sergeyevich became angry "You give him at least a Major". Malinovskiy for a long time did not agree, but Khrushchev insisted and Gagarin became a Major that very day.
From here on, I'll be referencing modern Russian media sources and some specials on Gagarin that I was able to find with a quick search on the TV studio website of RosKosmos: http://tvroscosmos.ru/?page=gagarin
After that, a massive reception was organized in Moscow. Gagarin received multiple awards and medals. There was a lot of media coverage of the events and he became a celebrity in the USSR. He went on an international tour, starting with a trip to Czechoslovakia. He took an ordinary flight where he was recognized and mobbed by autograph seekers. The pilot allowed him to enter the cockpit and steer the plane.
So, he became a celebrity in the USSR fairly quickly. As I mentioned above, the USSR was a pretty monolithic entity with a fairly uniform distribution of opinions and viewpoints supported by a state-run media. Gagarin was universally loved and the entire achievement was viewed very positively, especially with Gagarin as the face of the achievement.