You most certainly SHOULD worry about PH of soil. Soils can be acidic or basic and if you don't have your soil PH in an acceptable range, plant growth will suffer. If you know your soil PH is in an acceptable zone, then no, you don't have to worry about the PH of water so much, as the soil will buffer it.
While the PH of run off will not tell you the PH of your soil exactly, it WILL tell you where your soil is trending and provide useful information for you to act on if necessary. For example, say you water your soil with a PH of 6.5 and measure a PH runoff of 5 - that tells you something important - like maybe your soil PH is too low.
Most PH meters are junk, and most people don't keep up on calibration which makes them even worse. Believe it or not, the best way to check the PH of your irrigation water, and runoff water, or any other liquids you'd like to test, is to use the PH test kit strips or vial/PH drops - no calibration needed, no additional costs and accurate 100%.
If your PH is too high in soil, you do not want to water with a PH of 4.. You'd want to water with an appropriate PH of 6.0-6.5. Watering with a super low PH of 4 will just produce stress on the plant.
Lowering the PH of soil, if needed, is not an easy task and takes time - especially if it's in a pot and a plant is already growing in it. In your situation, you could top dress some elemental sulfur - I've never found a concrete ratio of how much to use per gallon of soil - but I'd probably start with a 1/2 teaspoon per gallon of soil, wait 2 weeks, test runoff and go from there.
There is so much misinformation to lead you astray, I with you the best of luck!