Yes a cold can make you feel weak and tired because when I have a cold I'm always tired and feel very weak until the cold has gone away.
A cold usually starts with sneezing, runny nose, Coughing and feeling tired.
It's very normal to feel weak and tired and have a fever, dry cough, a runny nose, chills, muscle aches, a bad headache, eye pain, and a sore throat when you have a cold and It usually takes less time to get over a cold than it does to get over the flu.
There's not much you can do for the cold but stay hydrated, eat things such as soups or chicken noodle soup, get plenty of rest and take some medication to help with the cold symptoms and in about a week you should be feeling better.