How to ascend for the ultimate reward?

To add onto Jutschge's answer, champs with high mobility will be extremely effective at both harassing and nabbing kills while the enemy team attempts to Ascend. It might be cynical of me to phrase it this way, but it is highly unlikely that a team will never have a chance to go for an Ascension.

If your team is lacking in one of the categories mentioned above by @Jutschge, fill it in. If not, then I would advocate for grabbing a champ who can control the field. Champs with the ability to quickly grab kills and get out reduce the number of opportunities for the enemy team to go for "Super Xerath". The list of these champs include: Nidalee, Elise, and Fiora. Each one is a high-end duelist, and can easily assist in teamfights as well as 1v1 fights, which can occur often with the fast-paced gameplay this mode offers.

The other type of champ that you might want to consider playing is a battlefield controller. On Summoner's Rift, these champs have all the potential when sieging or defending turrets. On Ascension, you can lock down entire areas of the map to assist your team. The champs I am listing, specifically, also have burst damage which is great for stealing Ascensions: Veigar, Ziggs, Twisted Fate. Veigar can instantly lock down enemies for your team to swoop in and kill, and Ziggs is phenomenal at dealing tons of AoE damage to enemies (with the issue of lacking true dueling potential versus assassins). Twisted Fate is half decent in this mode because of two facts: if he groups with his team and gets ahead, you can automatically kill any champ he stuns, and his ultimate will always mean that you can see where the enemies are, which is highly valued with champs like Akali and Shaco being more frequently chosen.