You can get by with card only. Sofia public transport (both the metro, as well as buses) is now contactless as well - just tap your card to ride! (turnstiles at metro stations, payment/card reader devices in bus). Most restaurants and shops will accept card payments as well - but just to be sure ask in advance if card payments are possible, as some smaller shops/restaurants might still accept only cash.

As for taxis, I really wouldn't recommend paying in euro. Why would a cabbie give you a honest market rate without a mark-up, if he needs to make the extra effort to get to an exchange office afterwards? You will for sure pay more than when you pay in Bulgarian leva. And that's not even taken into account that by asking if it's OK to pay in euro (which you have to do in advance, of course) you are giving yourself away as a tourist without any local knowledge - increasing the chances of being ripped off if you happen to encounter a dishonest taxi driver.

I'd still always suggest to carry at least a bit of the local currency in case you want to buy something small. Whether or not it's easier for you to take euros cash and change it locally, or just withdraw some Bulgarian leva from an ATM, depends on whether or not you are charged heavily for foreign card withdrawals (if based in Switzerland and using a Swiss card, I'd assume it's just as easy to just withdraw cash from an ATM - it's what I always do when travelling around Europe).