How do I check the DPI of an image for specifically A3 landscape print?

you'll need to know the original image's dimensions and the desired print size. Once you have this information, you can use the following formula to calculate the DPI:
DPI = (Total pixels) / (Print width in inches) / (Print height in inches)

Keep in mind that 300 DPI is generally considered the standard for high-quality prints, so if you want to achieve the best possible results, you'll need to make sure that the original image has a high enough resolution.

There are a few ways to determine the PPI. One of the simpler ones, since you're in Photoshop already, is to look at Image > Image Size. On your cropped image, type in the dimensions you want and the PPI will reflect it's new essential resolution: e.g. in my example I'm using the width of a vertical A3 (29.7) to see what my original 300ppi image would be at that width = 218-ish.

 In your case, type in 42 cm for width (landscape)

The DPI setting is part of the document properties and can be viewed in the file info or seen in the image resize dialog for instance. If you just cropped it from a larger image that had already correct values, that info doesn't change.

Mylenium 

In Photoshop, several ways to check what's currently set. Try Image Size for one, or you can set up this info to show in your Info Palette if you select the option. 

Does it matter; not really, no. This is simply a metadata tag. The number of pixels is key! 1000x1000 pixels at 300 dpi, 1000 dpi or 72 dpi is the same. It's 1000x1000 pixels. 

More here (an old but still valid primer):
http://digitaldog.net/files/Resolution.pdf