Park in the car park on the southeast side of Skeldergate Bridge. Walk back to the road and turn left across the bridge. Just after the bridge is a road junction on the right and there, at Baile Hill, are steps up to the walls.. Go up the walls there and walk round clockwise, admiring the view. There's a little museum in one of the bars (gates, not pubs) you pass through. Shortly after you pass the railway station on the left the walls turn right and you get the grandest view in York, the Minster over the rooftops, ahead and to your right.
Stay on the walls until you have to leave them at Lendal Bridge. Keep going in the same direction, across the river, then take the first right down Lendal. When you get to St Helen's Square, with Betty's Cafe on the opposite left corner, turn left down Stonegate. Admire the old buildings - you're walking through centuries-old streets. Right at the end of Stonegate is the Minster. Go in or walk round, as you prefer, then backtrack to High Petergate and walk southeast as it becomes Low Petergate. When the narrow street opens up into a small paved area, pass the chocolate story on your right then do a little right-then-left jink down The Shambles. At the end of The Shambles turn left past the little church, then left again to see Whip-ma-whop-ma-gate, the smallest street in York.
Turn back, passing the end of The Shambles on your right, and walk along Pavement then Coppergate. Turn left down Coppergate Walk, and you'll pass Jorvik on the right. Continue towards Fenwick's and you'll see a little ginnel to the right of the store: take that, and it'll bring you into the car park by Clifford's Tower and then to the Castle Museum. Go through the car park and between Clifford's Tower and the museum, turn left on Tower Street and the car park entrance is ahead of you.
There's a lot more to ancient York - a market, lots of old buildings, tiny alleys - but this will let you see a good deal of it with opportunities to stop at the bits you find interesting on the way.