Hello chetanshettt. Welcome to the Apple Discussions!
I find that my Airport Express wireless signal does not reach the far end of the next room, which should be about 100 feet away. Is this a problem that can be fixed?
The advertised range of the 802.11g AirPort Express Base Station (AX) is 50' for 54 Mbps; 150' for 11 Mbps and is based on an area with open space and no forms of Wi-Fi interference. It is possible that building construction (walls, framework, etc.), portable phones, microwave ovens, and other Wi-Fis in the local area may be the cause of a poor signal that you are experiencing. I suggest try changing radio channels until you find one that gives you the best performance.
Can I upgrade the software to 802.11n or does it require a new Airport Express
Sorry, but no, you cannot "upgrade" the 802.11g AX. For 802.11n, you would need to get the newer 802.11n version.