A 12-year-old TV will probably have an HDMI socket. (If it's 'HD ready' then it certainly does).
However the best way to watch Youtube videos on a TV set is to either get a 'smart' TV (with a Youtube app included) or a set-top box that makes your TV 'smart'. I recently purchased a Now TV box, which turns any TV into a 'smart' one. (It's designed for TVs with an HDMI socket but, through the purchase of a special lead, it can also be used with older TVs). It's got lots of apps (e.g. for 'catchup' TV services), including one for Youtube. The box on it's own costs just £9.99 but you can also add on subscription services (without taking out any form of contract). A one month subscription to the 'entertainment package', giving access to channels such as GOLD, Sky 1, Sky Arts 1, Comedy Central, etc, costs £6.99 but you can get the box and 3 months access for £20 or less in supermarkets. (Asda have had the boxes, with the entertainment packages, on offer for £16 recently - that's where I bought mine. Tesco charge £20 but give you a total of £2.70 back by way of Clubcard points)