Is it common to have teeth ache after eating or drinking something cold?

Common butnot normal: Tooth sensitivity is a sign that something has changed. It can be gum recession. Decay, a defective filling or crown, a broken tooth, a bad bite. You should see a dentist, you can try desensitizing toothpastes like sensodyne pronamel or mi paste. Tooth sensitivity is not normal, but is common, .

Slightly: It is common to have a toothache if drinking something very cold for a sec. If the pain lingers for a few seconds that is not common and may signal something is wrong. Also if you have some root exposure (your gums have receded) it will make your teeth a lot more sensitive.