You have an old "BX cable" installation. This was common 100 years ago they ran "flexible armored cable" in homes - quite often in conjunction with "knob and tube" wiring. There were NO separate "safety grounds" at the time. The panels DID NOT HAVE a separate "safety ground" bus panel. "Safety grounding" was SOMETIMES done at the meter panel, then a metal tube ran from the meter to the breaker box, supposedly providing a ground path, and then the supposedly grounded panel box itself provided a ground through the BX armor on the wires assuming the correct metal clamps were used on the jacket of the BX cable where it entered into the panel. Sometimes they would run a "ground" wire from the panel to a cold water pipe.
The CORRECT WAY to have electrical outlets in this setup was to have UNGROUNDED outlets WITHOUT a ground prong. Of course this would require using the "cheater" 3-into-2 with a dangling ground wire things everywhere. But at least the user plugging anything in would know the system was ungrounded and could either buy double-insulated appliances or take their chances.
"back in the day" people would sometimes drill holes in these panels and install a separate ground bus and ground wires assuming space existed in the breaker panel. Yours certainly does not have enough space for this IMHO. I personally would question if all those breakers in there would even trip at all they are so old. Plus you do NOT have a proper main disconnect breaker.
If this was mine I would demolish the main panel and buy a modern one with modern safety "smart" breakers in it. Your incoming cabling from the meter also looks like possibly a 100amp (or maybe lower) service and I would suspect the thing is already overloaded so demolish all that and replace with a proper modern feed from the meter. Those old 2 wire systems are safety hazards and at the least, doing it this way will give you some GFCI/AFCI protection and with an old wiring system like that, plus given the tampering it has likely suffered over the years, that would be the wise choice.
Look at it this way - you are already saving a ton of money installing your own mini-split, well I would rather be safe from electrocution in my own home than cool so take that savings and spend it on a main panel replacement. Yes I know you are looking for a quick answer here but there is none. There is NO safe way of "modifying" this installation to make it safer with ground wires and whatnot and the second you touch the panel like that you are required by law to follow code and there's no jurisdiction out there that would allow a panel like this to remain and meet code.
