well, i have a K&N style intake filter that bolts directly onto a SR MAF and I remember hearing that the 240sx uses the same one? don’t know for sure if its true though. and if your in a clean area with low wind, don’t worry about running the car for a while wihtout a filter. mechanics do it all the time and you never know about it. if the air is fairly clean then nothings getting in there anyway. or throw a rag or old shirt or something over the inlet and fasten down with a hose clamp or something if your unsure. not a permanent fix, but a fine temporary one.
As far as your problem, it really sounds like the starter solonoid. Without knowing more, that’s my best guess, on many kinds of solonoids you can pop them open(ussualy 2 screw) and see if the contacts inside are worn out or burnt. but some kinds of solonoids are pressed on and don’t come off. it’s also possible there is a starter relay somewhere that’s failing? but my first guess would be the solonoid. if it’s easy to take off the starter, I would try that, or take it to someone else to check it. the part should be around $40-50.