rpt007: 1. Generally it is no good idea to use RX or TX pins in an Arduino project if it is not necessary. TX/RX are used for serial communication and while loading a sketch up to the MCU; if you have other pins available, stay away from the RX/TX pins (you may modify your sketch accordingly)
Outages of 15 to 20 seconds are way too much being bridged by a capacitor of a reasonable size. Depending on the overall power consumption of your project, which imho will be around 50mA and up, you can calculate the time that a capacitor of 4700uF can bridge to go from 5V to 4.32V (this is the absolute minimum to operate an Arduino by a reduced speed (MHz)). This calculation will be in the magnitude of ms(!) and not at all close to only one second!
Hence the only advice I have for you: go with a chargeable battery to bridge the outages. In reply #2 I included a simple circuit for using the wall adapter to charge the battery while the mains is available and the battery taking over for the period of time when mains is gone. Think of a small capacitor (1nF .. 100nF) to avoid spikes being produced by your mains when it goes on or off - the value of the capacitor - this is something you should experiment with, as I have no clue what kind of spikes your mains is causing for you project.
I have not used Tx\Rx pin in this project. I am talking about the LED's for Tx and Rx on arduino. Suddenly Tx LED on arduino stops blinking and my internet connectivity goes off however Rx LED and others are working fine as shown in pic#1. And after unplug and plug the power cable to arduino, continue to working fine.
I have tried by connecting 2200uF capacitor as you can see in pic#2 but no success.
I will try with the chargeable battery to resolve this problem.