You can rent a trencher for about $80 for a day, that is what I did recently when I had to dig about 200 ft of trench through rocky clay soil. Or you could just hand trench in sections using a trenching shovel maybe 20 feet per day so you don't wear yourself out.

I would recommend going deeper then right under the sod, you may want to core aerate someday and you don't want to worry about poking holes in your pipe. I made sure the top of all my pipe was at least 6" below the ground so I didn't have to worry about it.

Also - since you are going through the trouble of digging don't bother with hose - animals could chew through it in no time and you will have to do it all over again. I would recommend getting some poly irrigation pipe, get 3/4" for good flow, and getting adaptors on both sides to hose thread. All available at home centers.

Also think about draining for the winter if you are in an area where the ground freezes. Best case - one end is the low point so that you can open the valve at that end and let it drain. Worse case - you could put a valve box in the middle where the low point is and put a drain valve there. Or if you have a compressor get the fittings to blow it out at the end of the season.

if you can save the sod and replace later, great. If not don't worry about it, if you overseed and water it correctly it should fill in within months and the next season you won't know where the trench was.