No if you use Solar Beam first turn, and your opponent uses Volt Switch and they switch in a Pokémon who has Drought, Solar Beam will not activate the second the Pokémon that has Drought switches in, if the Pokémon that uses Solar Beam moves first.
Source: http://replay.pokemonshowdown.com/ubers-96196032
However if you use Solar Beam first turn, and your opponent uses Volt Switch and they switch in a Pokémon who has Drought, Solar Beam will activate the second the Pokémon that has Drought switches in, if the Pokémon that uses Solar Beam is slower than the Pokémon that used Volt Switch.
Source: http://replay.pokemonshowdown.com/ubers-96189423
Hope I helped :)