It is a good idea to stay at Manali, a beautiful place, and also good for acclimatization. I think buses are limited on that route, better if you can rent a bike if you are a good biker, but don't if you are not, because the road is very very bad. There are number of water streams that are up to 1 feet deep. Better to opt for bus. Buses run from Kullu via Manali, and only in this summer season, in winter the Rohtang pass connecting Spiti gets snow covered.
Generally buses run in the morning time, because the streams get swelled with the higher discharge in the noon due to sun heat and therefore melting.
I will call someone to know the exact bus timings, wait for my message, I hope you are not in a hurry to know this.
RE: My Upcoming Solo Travel to Himalaya mountains in India in 10 days but I am having mixed feeling