Was Russian important to know for Eastern Bloc citizens, similar to how English is important to many countries now?

It was definitely important, but definitely not to the extent of English now. I'm speaking here about the experiences in Romania. For the Slavic countries, it may have been different.

In Romania, most students did study Russian in school, but far from everyone, there were other languages studied (particularly French, German, English and Spanish). In Romania, French and Spanish (and to some extent, English) are easier to learn due to the Romance origin of Romanian, which may explain their popularity, even back then. Nowadays 95% study English, with most common second foreign language being French.

The reason why learning other languages was encouraged may have been Ceaușescu's "independence from the USSR" policy: he wanted to trade with Africa (hence the need of English and French) and South America (hence Spanish).

Between 1965 and 1971, Romania was culturally very open to the West: for instance, during that period most movies at the movie theatres were American, most songs at the radio were in English-language: even many local bands sang in English.

In 1971, Ceaușescu started his own cultural revolution and he clamped down on foreign culture, particularly English-language: only Romanian and Eastern bloc languages (and later, some Italian and French) were allowed in broadcasts.

Of course, the American and British pop culture continued to be popular, through underground music sharing (that was the era of the reel-to-reel tapes). People listened to Radio Luxembourg's English broadcasts.

Overall, Russian may have been useful for trade within the Eastern Bloc, but English was already an important language to use for trade outside of it. However, culturally, English was far for important even then.