You may want to consider using key authentication as opposed to a password.
People will say use a password in addition to the key, but if your alternative is storing the password on your PC in a file anyway, someone with access to your machine owns you in either case.. So you just need to generate the keys. The requirement is: no-one but you has access to that key file.
http://www.linuxproblem.org/art_9.html
I'm in the same boat, have to use Windows, but for me www.mingw.org which gives you a shell, and the basic *nix tools - extremely useful for SSH, connect to remote Linux VPS, etc.. Cygwin, of course which is similar, and has an easier tool (setup.exe if I recall) to install new apps. I actually use git-bash with is mingw with git. No-GUIs. I've found this easy to just drop to the mingw shell when I need to use ssh openssl cut awk etc..
So running any remote command using SSH from the command line without third-party programs like Putty, or those with GUIs, etc.. Using the key authentication and offing password auth completely in ssh on the remote device (at least on devices where you have control) is some additional lockdown for the remote device, especially if you're the only one need access it.
Which leaves, scheduling the script. There should be a way to do that via batch file and Windows or within the command line environment.