Você pode usar Shift + Ins para colar o texto.
De PuTTY documentação :
Pasting is done using the right button (or the middle mouse button, if you have a three-button mouse and have set it up; see section 4.11.2). (Pressing Shift-Ins, or selecting ‘Paste’ from the Ctrl+right-click context menu, have the same effect.) When you click the right mouse button, PuTTY will read whatever is in the Windows clipboard and paste it into your session, exactly as if it had been typed at the keyboard.