First you need to make sure you have the right chuck setup. If the pump you have is the one in the link then it claims to be auto-converting between Schrader and Presta -- you can skip to the next paragraph. But if you have a slightly different pump it may require manual conversion. As you press the chuck onto the valve you should feel some resistance as the rubber gasket in the chuck snugly fits around the valve stem. If the fit is not snug the chuck probably requires manual conversion and is probably set up for Schrader, so you need to unscrew the knurled nut on the very end of the chuck and reverse the rubber gasket inside (taking care to no lose any spring or washer behind the gasket).
Regardless of whether the chuck is auto-converting you may need to tighten the knurled nut on the open end of the chuck, to make it tight enough that you feel some resistance from the gasket as you slide the chuck on.
But before installing the chuck you must unscrew the little knurled knob on the top of the Presta valve. Unscrew it all the way until it stops against the crimp on the top of the threaded rod. Just to be sure you have it open, press the end of the rod with your finger and listen for the puff of air.
Press the chuck onto the valve. If the Presta valve does not have a nut retaining it to keep it from sliding back into the tire you will likely need to use your thumb on the tire, on the back side of the valve, to keep the valve from sliding into the tire.
When you get the chuck all the way onto the valve (about 3/4" or 2cm) then flip up the lever to lock the chuck on. (Note that there have been a few chuck designs where you flip the lever up while applying/removing the chuck, and flip it down to latch instead of vice-versa.)
Once you get the chuck on properly you should be able to read the air pressure (if any) already in the tire. If not the valve is not open and you will likely not be able to get air into the tire.
Once inflated, remove the chuck and screw the knurled knob down snug (but not over-tight). And install the cap, if you have one. This helps prevent damage to the relatively fragile Presta valve mechanism and prevents accidental release of air.
You need to add air to your tires much more often than monthly. Once a week at the very least, and in most cases 2-3 times a week.