I will be arriving in London Heathrow airport from terminal 3, and after 8 hours departing from terminal 5. Can I walk to terminal 2 arrivals and use the paid lounge there without a UK visa?
You cannot use an arrivals lounge without passing through immigration. You will probably need a UK visa to pass through immigration, although in some circumstances visas for other countries are acceptable for a same-day landside transit. See the UK's official webpage but ignore the large text on the last page and read the full answer beneath if you have a visa for USA, Canada, Australia, New Zealand, Ireland, the EU (category D) or Switzerland.
I would advise you to use the lounges in terminal 5 instead, which are generally more pleasant than the landside lounges. The arrivals lounges are not intended to be pleasant places to wait eight hours before your next flight but are intended to be somewhere to have breakfast and freshen up quickly before a long day of meetings in London.
There are now two paid lounges in Terminal 5, the Aspire Lounge and the Plaza Premium lounge. The Aspire Lounge in T5 airside is the same brand who runs the paid landside lounge in T2. I think everyone agrees that the Plaza Premium lounge is better.
Review of the Aspire Lounge. / Priority Pass accepted / £40 / £20 surcharge for showers.
Review of the Plaza Premium lounge. / £40 / Free entry for Amex Plat holders / £15 surcharge for showers.