You can see the cup has wrench flats. If you can get a wrench in there without removing the drive side crankset you could probably tighten it down with a large adjustable wrench. Keep in mind on this frame the drive side cup will have left-hand threads, so go counter-clockwise to tighten.

However based on the photos the drivetrain does really need a disassembly, cleaning, re-packing with grease, and re-assembly. For this you will need a crank puller to pull the cranks off the bottom bracket spindle, and a hooked spanner for removing the non-driveside bottom bracket lockring. Both bicycle-specific tools unfortunately.

If you don't do this, you run the risk of ruining your bottom bracket bearings and cups, which will be a more expensive repair than simple disassembly and cleaning of your existing BB. If there is a bike coop nearby you can borrow the tools mentioned above and look up the Park Tools video on bottom bracket install and removal. Otherwise please bring it to a bike shop for service.

On a separate note, there is some rust on the frame that you should remove. And PLEASE spray some lubricant into the bottle bosses on the downtube and plug them with the appropriate M5 or M6 bolt. Letting the internal and external rust continue will eventually cause the tubes to fracture which could result in a catastrophic road accident.
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