Yes, if a gust of wind was blowing directly into a static port, it could cause a temporary increase in measured static pressure and a drop in the altitude readout.
Traditionally this was corrected by putting static ports on both sides of the aircraft and connecting both of them together, in a T layout. This reads the average pressure on both sides.
Modern large aircraft instead use digital sensors on both sides and the air data computers combine the data. This eliminates potential leakage in the piping, which has caused gross altimeter errors in the past.
Large aircraft combine data from multiple sensors, particularly accelerometers from the inertial nav system, and AOA and sideslip probes. These allow corrections to be made for gusts and local airflow, important to be able to meet altimetry accuracy requirements for RVSM and Baro-VNAV. Another trick is that at least one sensor will be placed asymmetrically, so that vertical gusts will have a unique signature.
This is why air data functions are in the same box as the inertial reference units, forming the integrated air data inertial reference system (ADIRS).