If my sinus problems where indeed caused by allergies, would anti-histamines definitely have some effect on it?

Sinus infection. It is common to get sinus infections if one has allergies that are not controlled. There are other causes of sinus infections as well. If recurrent sinus infections are secondary to allergies than controlling allergies with antihistamines or other medications is needed.
Maybe. If the trigger is allergies and the allergic trigger can not be avoided, the proper anti- allergy medication should help.