Japan is slowly coming to accept credit cards, but it is still largely a cash society, and that cash is yen. If you need to exchange yen into a foreign currency or vice versa there are a number of options for you to employ. You can buy yen at some banks, major post offices, international airports, and in some limited cases money exchange is available in hotels and department stores.
Buying yen with major currencies is pretty straightforward, but buying foreign currencies with yen is a little more complex. Most of these options simply do not keep a large amount of foreign reserves on hand. In practice, this means you can get your currency exchanged at many places, but most will ask you to come back in a week to pick up your money. If you can wait, or require a fairly small amount of cash exchanged, there is a fairly exhaustive list of providers that may be of assistance through the link below.
If you are in a rush, or have a large amount of cash to exchange your options are very limited. The best place to go to in Nagoya to foreign currencies is an MUFJ branch with a a dedicated foreign exchange desk.
Bank of Tokyo-Mitsubishi UFJ Nagoya Chuo Shiten
Hirokojidori is the main street through Sakae and Fushimi. Going from Sakae towards Fushimi this branch is on the right side of the road. This branch is the main foreign exchange branch for UFJ in Nagoya. It is most likely to have cash, and larger amounts of foreign cash on hand than any other branch.
Google Map http://goo.gl/maps/Q2aPg
Bank of Tokyo-Mitsubishi UFJ Sakaemachi Branch
The Sakaemachi branch is located on B1 of a building near Skyle and Melsa. It has a dedicated foreign exchange desk, but if you need large amounts of cash the Nagoya Chuo Shiten above is a better bet.
Google Map https://goo.gl/maps/gDxRH
These private companies also exchange foreign currencies. If you need your cash on a weekend, these may be your only option besides the airport. These will cost you more money than other options.
Ticket Station Sakae
Goggle Map http://goo.gl/maps/KRK9x
Travelex Japan Nagoya Sakae
Google Map http://goo.gl/maps/ZekAl