It should be possible for free. I'll need one of the other experts to help me on this as I don't have any network experience.
First, to gain access to the machines - Use some remote access software. VNC works well and there is a simple free version. I'm wondering if a free version is going to be secure another for what you need.
http://www.realvnc.com/download.html
Now for the printnig. Are these two offices on the same network? if so it would be pretty easy. You would just have to add a new printer on the machine you are accessing, give it the network share of the printers you want the documents to come out of (e.g. //OfficeA/FirstFloor/Printer). Then it would just be a case of going to whatever program you need to print from, press print, choose this new printer and you're done.
If you have some network admins in your office, have a chat with them. They might be able to help you with the printer sharing business.