var req = { mandrill_events: '[{"event":"inbound","ts":1426249238}]' }

console.log(Object.keys(req)[0]);

Make any Object array (req), then simply do Object.keys(req)[0] to pick the first key in the Object array.

To answer your titular question, you use [0] to access the first element, but as it stands mandrill_events contains a string not an array, so mandrill_events[0] will just get you the first character, '['.

So either correct your source to:

var req = { mandrill_events: [{"event":"inbound","ts":1426249238}] };

and then req.mandrill_events[0], or if you're stuck with it being a string, parse the JSON the string contains:

var req = { mandrill_events: '[{"event":"inbound","ts":1426249238}]' };
var mandrill_events = JSON.parse(req.mandrill_events);
var result = mandrill_events[0];

the event property seems to be string first you have to parse it to json :

 var req = { mandrill_events: '[{"event":"inbound","ts":1426249238}]' };
 var event = JSON.parse(req.mandrill_events);
 var ts =  event[0].ts