Should I replace my fridge for power savings?
To give an example, in Europe with costs of 0.3 Euro per kWh, a new fridge (size of a dishwasher or washing machine, price starting from ca. 150 Euro) will pay off in under a year, if the fridge is older then 15 years. 

The insulation will be much better, but since there is some space around the old fridge, glueing polystyrol foam insulation panels to the left, right and upper side and to the doors would reduce the energy consumption - and the noise level when the compressor is running.

Otherwise there should be a forum or test in your market to find a new energy saving model which is reliable and not expensive. 

And what could also help to reduce the energy bill: installing a small balcony photovoltaic system. Depending on the subsidiaries, procurement and feed -in and -out costs, it will save money after some years, especially if the local code allows simple plug-in systems. During day time, it could harvest some  energy for the fridge, the router, computers, TVs etc.
Again an example from Europe: a 2 panel system (peak power ca. 600W) can pay off in under 5 years without subsidiaries and without any feed-in payments, if the installation is done by DIY. Some owners decide to feed-in the energy for free that is exceeding the own consumption, since the burocratic efforts/complicated tax declarations and additional meter rents are not paying off.

And in case of power outages/emergencies photovoltaic systems are a reliable daytime electric power source which should supply enough power to run a cell phone loader or laptop or receiver or TV, even in cloudy weather.