I'm not familiar with monogame but LogCat can be viewed not only by using apps but also from command line tool adb(android debug bridge).
Assuming that you had to install android sdk You should be able to do this steps in your command line/shell:
- Change dir to /YOUR_SDK_PATH/platform-tools eg.
cd ~/android-sdk/platform-tools - Run adb with logcat option
adb logcatfor windowsadb.exe logcat
The problematic outcome is that it prints whole LogCat output so you may want to filter it for your needs.
For more information about filtering check this SO threadcheck this SO thread.
Info: You may need to enable option 'USB Debugging' in 'Developers Options' on your device.
If command line related solutions aren't for you try to install Android Studio. It has 'Android Monitor' panel that does the most work and allows you to define own filters using GUI.
Extra: After some googling I've found tool called mLogcat which maybe be useful for You as alternative to Android Studio.