Russian oligarch Alexander Lebedev has always sung his own tune. Now the billionaire media magnate has jumped into the ring with anti-Putin activist and blogger Alexey Navalny, just as the Kremlin tightens the screws on dissenting voices on the internet.

Lebedev said on Wednesday that he would present Vladimir Putin, Russia's powerful president, with a new debit card designed to fight corruption. The gesture is an ironic one in a country where accusations of corruption and the accumulation of untold wealth continue to dog Putin and his close associates.

The card will be put out by Lebedev's National Reserve Bank and will put 1% of all purchases toward RosPil, a fund launched by Navalny to expose corruption inside the Russian government.

"I'd like to 'marry' this project with the authorities' official position, so that on the official level, the authorities support citizens' intention to finance the struggle against corruption themselves," Lebedev said in a statement posted on his website.

He added that he wanted "to approach Vladimir Vladimirovich [Putin] as a citizen, so he shows us all by example."

Navalny has risen to become one of the leading opposition figures inside Russia by focusing on exposing government corruption via his popular blog.

But under a new law, his internet presence – and that of millions of Russians – is under threat, critics warn.

The Russian parliament unanimously adopted a controversial bill on Wednesday that will create a federal website "no" list that would force site owners and internet operators to shut down any site put on it. The bill's supporters say it is designed to crack down on child pornography, as well as sites that promote drug use and teenage suicide. Its critics warn that it will lead to the creation of an analogue of China's "Great Firewall" inside a country that has seen the internet flourish as a platform for free debate.

Russia's leading websites – the search engine Yandex, the blogging platform LiveJournal, and the Russian-language Wikipedia – had joined together to warn against adopting the bill.

The Duma, Russia's parliament, appeared to take some of the concerns of the bill's critics into account, but analysts warned that dangers still remained.

MPs removed vague language that would allow any website with "bad content" to be shut down, according to Duma deputy Ilya Ponomaryov, an opposition activist who has come out to support the bill. Under the new language, authorities will only be allowed to shut down sites that contain child pornography and drug and suicide promotion. All other sites would require a court order before being placed on the blacklist.

Yet Russia's court system is notoriously corrupt and pliant – less than 1% of cases brought before Russian courts result in a non-guilty verdict.

And critics still warned that the existent language left room for manipulation. Those opposed to Putin have long feared a crackdown on the internet, where opposition discussion and protest organisation has flourished amid a largely state-owned media environment.

"The proposed methods provide a means for possible abuse and raise numerous questions from the side of users and representatives of internet companies," Yelena Kolmanovskaya, the chief editor of Yandex, wrote in a statement posted on the website on Wednesday.

Andrei Soldatov, an expert on Russia's security services, said that in the wake of the mass protests that hit the country as Putin prepared to return to the presidency late last year, his sources inside the Federal Security Service (FSB) became particularly concerned about the power of social networking sites like Facebook.

Once a believer that Russia's internet would remain free, Soldatov has joined the legion of critics who believe a larger crackdown is in store.

"Before this law, we've seen a lot of cases, mostly in the regions, where local prosecutors requested local internet service providers to block access to certain sites," he said. "Now we got a nationwide initiative – it's a completely different picture."

The bill's supporters tried to brush off opposition to the law by accusing its critics of paedophilia. Yelena Mizulina, the bill's co-author, said: "The online community initiated the need for adopting this law themselves, that's why I'm sure not all of the online community is against it – just certain circles that can be associated with the paedophilia lobby," she said, the Interfax news agency reported.

The bill comes amid what activists have called an unprecedented post-Soviet crackdown on opposition activity.

Also on Wednesday, the Duma gave initial approval to a new libel law that would introduce fines of up to 500,000 rubles (£9,850) and sentences of up to five years in prison. Government critics warn that the bill is another means of crushing dissent.

The Duma is expected to adopt the law by Friday, as well as a new law that would require nongovernmental organisations funded by foreign grants to advertise themselves as "foreign agents".