Gillybob From nearish to Waverley, two buses go to the Botanic Garden - a number 27 from The Mound or Hanover Street or a number 8 from North Bridge.
If you search at www.travelinescotland.com/?lang=en putting in to arrive at the Botanics by 11 on a Monday, you will see that the 8.29 from Newcastle gets you into Waverley at 10.20, and sends you to the stop on the Mound for the number 27 ( I would go to Hanover Street myself - not so hilly! ) to arrive at the Botanics at 10 to 11.
If you put into it a 10.00 arrival at the Botanics, it gives you an earlier train and sends you onto North Bridge for the number 8, which gets you to the Botanics at about 10 to 10.
Either way, you need your bus pass or £1.60 ready in your hand - they don't give change!