...the signal is getting weaker especially toward the end of the day...
This may be caused by other people in your local area with wireless routers interfering with your signal. You should try changing the channel your WiFi is transmitting on to see if this improves your signal.
Can I install router without plugging the LAN cable to my PC?
That depends on the make and model of router.
Some will have the wireless enabled by default with no encryption, some will have the ability to link your computer to the WiFi with encryption by pressing a "WPS" button on the router, and some will have the wireless completely disabled.
Either way it's safest to link the router up to the computer for configuration - at least for the WiFi configuration portion. If you configure the WiFi next to your computer, then move the router to its final destination once you are happy with the WiFi configuration you can then proceed with the Internet configuration remotely.