The programme will begin on Monday and is expected to last until the end of November.
The scaffolding will be in place around two sides of the building but the landmark will remain open to the public while the work is being carried out, although the observation post will be closed throughout.
New measures will also be installed to “bird proof” the castle.
Historic Environment Scotland (HES) said spikes had already been installed across a number of windows but the height of spike needed to deter seagulls and pigeons meant this option was not considered appropriate for observation post windows.
Instead, bird-proof netting will be put up.
Broughty Ferry Liberal Democrat councillor Craig Duncan said: “The most visible legacy of the works will be the installation of bird-proof netting around the windows of the observation post.
“Once the work is complete, HES has indicated that visitors will be able to see the netting as they enter the room.
“They have assured us that the netting will not prevent visitors from enjoying the view, with or without binoculars, nor from taking photographs. The netting across the observation post windows should also allow a regular regime of window cleaning, significantly improving the experience for visitors.”