If you plan to install a set of new rings than yes you will need to first cut the ridge out of the bores, clean the short block really well' then hone the bores so the new rings will seal a budget ball type hone will do the job. The pistions will need to be cleaned of carbon deposits and the ring groves will need to have the carbon scrapped out with a ring grove cleaner tool so the new rings can fit properly, plus a Torque Wrench for reassembly of the engine. X2 usually most of these tools can be borrowed from one of your local auto parts stores.
To accolish an overhaul of the short block for a very low cost get a rering kit, Falcon Gobial has a good quality kit for $115.00 (plus tax in some states) with free shipping (See the below link). The kit comes with new rings, rod bearings, and a full gasket set, depending on the condition of the chrankshaft mains you could also add those bearings. Add to this a new set of freeze plugs for the short block and head, and maybe a new timing chain set if there's any slop in yours then yes replace it for sure, you will maybe be at about $200.00 to $300.00 for parts and some of the optional items below for the compleated short block plus your time. Add to that a valve job (check your local machine shops for their prices) and a .025 surface cut on head (to compensate for the thicker head gaskets) and your done with the long block overhaul. However if your really short on funds than if the heads valve seats, guides, and valves look to be in fairly good condition you might be able to get by with just hand lapping them yourself this is very inexpensive.
Again If your budget is really streached to the limit these items are also Optional first would be a hot tank or Jet wash of block, head, and other parts, usally not that much when included with other work shuch as a valve job and well worth this cost to get everthing really clean, cam bearings need to be replaced if the block is hot tanked but not always nessisary if it's Jet washed with the camshaft left in the block, last a can or two of spray paint if you want to also detail the engine while you have it out.
After everthing is dissasembled and parts are checked for their condition and cleaned you can order your ReRing Kit. Keep the lifters in the order they came out of the block unless you are planing to also replace the cam and lifters too. Check the camshaft and lifters bottoms for wear replace if they are worn. A stock reground cam (cheapest option) plus a new set of lifters will probally add $100.00 to $150.00, hopefully you might be ok in their condition. There are also performance cams available from Vintage Inlines our site sponsor. Look at the tops of the pistions for oversize markings so you can get the correct sizes. No markings is a Standard size, .030, .040, .060 ect. are oversizes. Also check the back of rod and main bearings for markings again no mark, FoMoCo in oval, or Std. Is standard, .010, .020, .030 ect. are undersized. Since you have good oil pressure your oil is probably in good usable condition or you could replace it with a new one. That should just about do the compleate long block engine overhaul. Since your engine runs and has good oil pressure, good compression except for the number one cylinder being weak I have a hunch that you can get by without a lot of expense replacing only the very worn parts and some minimum amount of work. Your doing a great job on your little 1962 Falcon keep it up best of luck
Ford 144 six ReRing Kit
https://www.ebay.com/itm/Ford-rering-en ... xySy9SQbUX