Start from the forestry entrance at R90294 66782 A and walk uphill, taking two lefts and one right off the main track onto a very rough track thats almost blocked near the end with felled trees, to R90538 68028 B. A rough but managable push to the left through the trees and undergrowth will bring you to open bog and the heathery summit cairn. Return the same way.
Another route is from Lisgariff East at R919 692 C.
The summit has views of the distinctive Shannon hill area scenery, smallish hills, forestry, farms and there is a view of Keeper Hill, the local mountain.
The construction of a broad track has made access (as of March 2023) to Cooneen a deal easier. Park at R90294 66782 A and proceed north for about 800m to a crossroads of forest tracks at R90127 67422 D and turn left. About 200m along you will see the broad track of reddish gravel on the right at R89948 67429 E, (presumably constructed for vehicular access) running in a straight line northwards, steepish in parts, but this will take you to within 250m of summit. At the track's highest point, cross the bank on the right to enter the heathery summit area. Watch for a narrow, single file trail as soon as you cross the bank, which makes the remaining trek through heather, sedge grass and reeds a great deal easier. This continues almost to the top, which is an elevated grassy mound, with some stone hints that it might be of megalithic origin. By this route, 35 minutes car to top. Linkback: https://mountainviews.ie/summit/558/comment/5317/

Ouch, ouch and ouch again! said the fool as he struggled through upended sideways spruce and burly briars. Its a wonder these days I have any skin left on my knees and shins. I started from the forestry entrance at R90294 66782 A and toddled along uphill to R90538 68028 B where a rough but managable push through the trees brought me out to open bog and the heathery summit cairn.A rest there for a few minutes taking in the views of the Shannon and North Tipperary hills and in the distance the Galtys,Knockmealdowns and Comeraghs. The North Tipperary Archaeological Inventory mentions this cairn

"Situated on the summit of Cooneen Hill with panoramic views in all dircections. A circumlar cairn (base diam. 8.7m N-S x 10.3m E-W; top diam. 5.1m) sitting on a natural knoll which continues to descend to S. The cairn (H 0.8-1.6m) is covered in moss and heather with small stones protruding. Kerbstones defining the top of the cairn at S. The interior is flat with a flat, with a slight drop below the encircling kerb. Large flat boulders in the interior may be lintels although three or four are loose."

A word of advice though, descend the way you ascended from the coordinates at the track end above lest you lose your epidermus as I did.

On the way down to your right you might make a short detour to R89750 67868 F to see the Foilnamuck wedge tomb.
Again the North Tipperary Archaeological Inventory says

"Situated on gently sloping boggy ground on the lower slopes of Cooneen Hill. The remains of a gallery, 4.6m long, survive. A septal-stone, partly embedded in a field-fence, marks the SSW end. Three contiguous stones remain on the more northerly side of the gallery. Beyond the easternmost of these there is a small set stone of uncertain fuction. There are four sidestones on the S side of the gallery. Two are alongside on the SSW and the other two, which are smaller examples, stand apart from each other towards the NNE. There is a thin slab outside the junction of the former pair. This supports one end of a roofstone the western orthostat on the N side of the gallery. Outside the latter orthosat there is another, apparently the lone survivor of a close-set outer wall. A decrease in gallery height and width towards the NNE is indicated. (De Valera and Ó Nualláin 1982, 82, No. 3)" Linkback: https://mountainviews.ie/summit/558/comment/4548/

My access to this mountain began from the forest entrance at Lisgariff East r919 692. You should allow an hour and a half for a return trip to the summit from here. It has wonderful views towards Keeper Hill. There is a large field on the shoulder (kinda Sound of Music-esque which makes for some marvellous fast descending if you have the energy. Like Knockadigeen and Ballincurra Cooneen is relatively firm underfoot (compared to recent trudges through Slieve Blooms) Linkback: https://mountainviews.ie/summit/558/comment/3029/

A view of plantation-scarred Cooneen Hill from plantation-scarred Ballincurra Linkback: https://mountainviews.ie/summit/558/comment/3030/

Knockane affords this view of Cooneen Hill from the SW. Linkback: https://mountainviews.ie/summit/558/comment/14993/