There's no chance to upgrade the OS. Mac OS X 10.7.5 (Lion) is as far as that Mac can go.
One of the big problems you're going to face is that Lion's version of Safari is not going to work with most modern https sites. After researchers discovered serious vulnerabilities in the security protocols originally used for https connections, the Web moved to newer protocols. Most Web servers offering pages over https (the 's' stands for "security against eavesdropping") will now insist on newer protocols that Lion's version of Safari does not speak.
There may be third-party Web browsers out there that will run on Lion, but I don't know offhand if any of them are recent enough to overcome this show-stopper problem. Even if they are, they will probably be far too outdated in other ways to let you do much Web browsing.
For instance, consider the Firefox browser. Mozilla is one of the better application developers as far as supporting old versions of the OS goes.
- The current version is 125.0.2 (64-bit). I believe it requires at least macOS 10.15.
- Firefox 115 ESR will run on macOS 10.12 through 10.14, but won't get any security updates after this September.
- Mozilla has already stopped providing any support for older versions of the OS, such as Mac OS X 10.7.5 (Lion).
Basically, don't count on doing any Web browsing using Lion or its predecessors. And if you can do some browsing, don't automatically assume that any https connections using old security protocols will be secure.