Let's make some assumptions about jungle Amumu:
- requires blue for an efficient jungle start
- can gank at lvl 2
- builds defensive and AP items
- deals primarily magic damage
The first option (as you mentioned) is to steal Amumu's blue. Amumu relies heavily on tantrum to clear his jungle quickly, but this can quickly consume his mana. This prevents him from ganking and drastically slows his early jungle. (note: With jungle camps now providing health/mana, Amumu can still manage the jungle without blue. In times long past, stealing his blue completely destroyed his first jungle clear).
The second option is to ward/counter-gank. Bandage toss gives Amumu a strong lvl 2 gank. With map awareness and a key ward, the opposing jungler can counter-gank Amumu early before he has any defensive items.
Some junglers also have strong skills when dueling Amumu. Mundo, Shyvanna, and Lee Sin will typically win duels without outside help, and these champions excel at counter-jungling him as well. This can be especially true since Amumu tends to jungle with low health early on.
As the game progresses, Amumu's ganks will rely on hitting with bandage toss. Keeping wards in common gank locations and using minions for cover minimize his chances to succeed. Purchasing MR can help, and having cleanse or a QSS help immensely with escapes. Amumu is also susceptible to silences/stuns, so a team with high CC can prevent these. Also, a Banshee's Veil is very effective vs Amumu.
Janna deserves a special mention, as her ult will directly counter Amumu's if he doesn't kill her first.
tl;dr: Amumu is a tanky jungler and great initiator. He can be shutdown the same as other junglers via counter-jungling and early invades. His strength comes through mid to late game, but if he doesn't succeed early, he'll never get there. Strong early-game junglers who constantly steal his jungle are the best counter to him.
This is just my opinion, but I hope that it will hold some insight for you.
In regards to the current meta and possible counters to it, there are no 100% counters.
One of the biggest variables in every game of League you play is the enemy teams skill/teamwork. No two teams play the same and no two players will respond to different situations in the same manner.
Many players have become accustomed to the current meta game of sending-
1 Bruiser type solo Top
1 AP caster/nuker/assassin Mid
1 Jungle
and
1 Ranged AD carry and 1 Support Bot
This team composition has been found to be not only the most efficient, but to provide in a typical game the highest chance for success.
This does not however mean it will always result in a win for the team that uses this format.
One of the reasons for this is that with set lane compositions, you often know who you will be facing in lane before you ever get there. This allows for strategies and tactics for each lane to be devised before you ever encounter your enemy in the lane.
Teams that decide to try something different, whether its sending that AP caster/nuker bot with your AD carry, not having a jungle so that your bruiser top is no longer going to get zoned out by that counter pick, sending your tank mid, all of these things can not only confuse the enemy team, but they can put them out of their comfort zone.
Being out of that comfort zone, not knowing how to deal with that dual top lane or that tank that you just can't seem to do anything to mid, can majorly screw with a player and a team's mindset.
Counters to the meta that I have seen be effective were not always necessarily fantastic teams that just switched up where a specific role goes. Part of the game of League of Legends is the mental aspect to it. A team that can confuse, agitate, or otherwise just throw off the shackles of what the enemy expects them to do, can win games regardless of filling those typical meta roles.
I have personally seen teams of tanks, teams of assassins, teams of AD carries and teams of champions that would typically be viewed as supports just roll the enemy team. Not because their team composition was better, but because they were able to change the flow of the game. They were able to force the enemy team to deal with situations that they normally would not find themselves in, fighting champions that they may very well not know how to play against.
The biggest thing with these teams, regardless of their team composition is that they are able to take control of the game. When they gained an advantage, they kept it. They push, gank, go for dragon/baron when they can take it, ward and communicate with their team. They "know" what they need to do to win, and they also know that they often times have to win before the game goes on to long.
In conclusion, although there is no "hard" counter to the current meta, with proper knowledge of the game, a little bit of innovation and a whole lot of teamwork, it is "possible" to win with something other than the current meta. (You just have to try!)
Best Answer
It's very rare that you'll have the opportunity to, with certainty, pick against a kill lane. Typically, they will be picked in response to your more traditional bottom lane choices (AD and a support). Moreover, if they picked one early on, it's very easy to confuse their choices with either a top or a jungle choice. In fact, they have the option of switching from a top/jungle choice at the last minute if they feel a kill lane would hurt your team.
That said, if you get the opportunity to pick against one, don't outpick yourself. You already know that they need to survive on harassing you until you're low, then killing you. Failing to do so doesn't optimally scale their champions (usually), and they're not built for farming. To pick against a kill lane, you have a couple of options:
In any event, the core point is that neither you or your support can start dying. As mentioned in a related question, "What is the idea behind a bot kill lane", the benefit of a kill lane is the indirect ability to leave their lane and assault other lanes or the jungle. Prevent them from killing you, and they have to remain in the lane.
If you've already picked your AD and support bottom, then get a kill lane, there are a few strategies you can employ to reduce their effectiveness. These apply in the above case, as well:
Once you have a lead, continue to play very cautiously. Like most top and jungle bruisers, many top lanes are dangerous even if you only start feeding them late into the game. Playing against them requires you to be conservative, as their lane is designed to punish over-aggressive play.
Finally, never be afraid of building defensively as the AD carry. If you cannot take the lead in your lane, at least you can not die.