I will convert to Freedom Units as that is what I am familiar with:
1028 millibars (the top of the spike) is 30.36 inHg.
1025 millibars (the minimum) is 30.27 inHg.
As a rule of thumb, at sea level, one hundredth of an inch of mercury corresponds to about ten feet of pressure altitude. Thus this difference would correspond to a difference of about 90 feet, if a pilot happened to set the altimeter at 2100 (the spike's maximum) and did not reset it by 2130 (the minimum).
This is a very noticeable difference, and would certainly trigger a "pressure rising/falling rapidly" remark on a weather report. But it really wouldn't have any detriment to most aircraft. Separation standards (both between two aircraft and between an aircraft and the ground) are quite conservative, and can handle a discrepancy of 90 feet.
It would be most dangerous for an aircraft on a non-precision approach, as there is no vertical guidance along the approach path and the pilot must instead make sure to not descend lower than a specified altitude at one or more specified points. An altimeter setting higher than it should be will cause the aircraft to be lower than expected ("high to low, look out below"). But again, procedures are designed with a margin of safety—a difference of 90 feet is probably not enough to cause an accident.
In the flight levels there would be no noticeable effect. Pilots and autopilots would silently adjust their altitudes to maintain the same pressure altitude, which would indeed result in a difference to their true altitudes, but because everyone uses the same altimeter setting everyone would change at once and separation would never be compromised.