Same approach as for O'Dell's Gully. The central gully line is visible as soon as you enter the ravine bowl, look for the straight 45 degrees snow slope in the middle of the headwall. Even if it is regarded as an easy climb, precautions should be taken as this route is in avalanche terrain and new snow loadings can cause slab instability. Always register at Pinkham Notch and read carefully the avalanche reports and forecasts.
A very nice route for someone's first "alpine" ascent. The slope is a sustained 40 to 50 degrees with a 50 degrees ice bulges near the end of the first pitch. Depending on the winter, this bulge might not be visible at all so don't expect to be able to put anything in it. Also, around half way in the first pitch, you will encounter an avalanche run out zone from Pinnacle gully. Some large ice chunks can detach from Pinnacle and end up in Central, so be on the look out. Depending on the thaw, you can expect to encounter some loose rocks on the last pitch as well as running water under the thinner slab. Some climbers having climbed other gullies in the day sometime descend using the central gully, so be on the look out for humain triggered debris. Having successfully climbed Central, you can descend using the Lion's Head trail via the Alpine Garden trail.
As for any snow climb, it depends highly on the snow conditions. Can be done unroped with mountaineering ice axe and walking crampons. However, be aware that avalanche conditions can exist in the gully and unstable slabs might encourage you to go roped with a couple of stakes for a running belay. Also, as stated in the route description part of this page, depending on the season and conditions, an ice section in the upper part of the route can be encountered. This section can either be tackled directly or bypassed to the right. Depending on the ice conditions, don't expect to be able to put screws in it and be prepared for a possible run out there.
Avalance forecasts and reports for Central Gully and the other Hungtinton ravine climbs can be found at: http://www.tuckerman.org
I would like to respectfully disagree with your characterization of the ice bulge on Central Gully. While it is true that there are times when the ice is indeed minimal or nonexistent, as you describe, I've seen it where there were actually several full pitches of ice. Also, while I agree that competent mountaineers can do it with just a single mountaineering axe and nontechnical crampons in average conditions, I would still recommend a second ice tool and at least several ice screws to a party attempting it as a first Huntington gully climb.
In summary, I would propose the following alternative description of the ice bulge: "The amount of ice can vary from none to a few meters to a full pitch or more, depending on snow amounts and other factors. In any case, the maximum difficulty of the ice is around WI2."