That packet of photocopied cards you received from the National Archives... It's not a pension, it's something called a Compiled Military Service Records. The blog Ancestry Insider offers a good history of the documents. In theory, your ancestor's CMSR should contain all the information about his service contained in military records. This would include a date of "muster-in," a date of muster-out (discharge),sick rolls, and more. (Sometimes you can find details elsewhere, such as a pension file or a court martial.)
So, if you know your ancestor served in a Connecticut unit, how do you get a copy of his CMSR? Actually, it's a little easier said than done... You can get basic information about your soldier on the Civil War Soldiers and Sailors database. Because it acts as an index for the CMSR, the site will give you a name, unit, and rank and little else. Fold3.com, the digital source for CMSR, is missing Connecticut units.
What are your remaining options? You can review the microfilm of CMSR records from the National Archives. The Soldiers and Sailors database will also list something called a "film number." That film number will give you the information you need to locate the correct microfilm. You can also order a paper copy of the file through their website (my preferred option). Happy research!