Actually, Charleston has a license to carry ordinance (Charleston City Code § 78-163)that requires a license to carry without distinction between open and concealed carry. This ordinance is identical to an oldstate law that was declared unconstitutional in State ex rel. City of Princeton v. Buckner, 180 W.Va. 457, 377 S.E.2d 139 (1988). The Legislature responded by changing the law to require a license only to conceal. However, Charleston's ordinance remains on the books in its old form but I do not believe anyone has ever sued to seek an injunction against its enforcement. However, if you have a concealed handgun license, this is a moot point.
Also remmeber that Charleston has a separate ordinance (Charleston City Code § 78-165)prohibiting carrying on all city-owned property (Dunbar and South Charleston have similar ordinances). These ordinances are all grandfathered under the current preemption law, W.Va. Code § 8-12-5a (1999). HB 3335 would preempt these and other grandfathered ordinances.
Although I cannot post direct links to the specific sections, you can find the applicable Charleston city ordinances here.