I swear....nothing ever happens until I go out of town. I had not heard from a single renter for six months (5 rentals). I go out of town with spotty reception and a limb falls from a tree and puts a fricking hole in a roof when three days of rain is scheduled, a toilet starts leaking, and an AC goes out all at three different houses.
I have a plumber, electrician and AC guys I trust and don't mind leaving those numbers with tenants. I emailed the tenants with the toilet and AC and gave them contact info and told them to schedule the work. The roof was problematic because I do not want to pay a roofer for an emergency call. Short term, I called my boss's college age son and paid him $100 to go over there on a Sunday, pick up the debris and secure a tarp over the hole and take good pictures. This is a kid I trust who has handy man-ish experience. Once the pics came in, I figured my local handyman could do it and I called him and gave him the tenant's number. We agreed with scope of work and price and everything over the phone.
I could've done most of this over email via the college kid and had him contact the handyman for me. I do not think I could've done everything without at least internet.
My advice would be to pay somebody X amount up front and Y amount for each incident to be "on call". It does not have to be an experienced handyman, it can even be a kid or your neighbor or whoever. You just want somebody good at problem solving/decision making to keep the house standing until you get back.