What would I need to open a circuit (a usb cable) for a few seconds, on a given time?
If you want a solution that is easily enough implemented [tm} without too much expert knowledge or experience you could use an off the shelf domestic event  timer and a relay.

(1) Time period generator

You can buy timers which are user settable and which count down to zero, sound an alarm for typically one minute then reset and start again. I have seen timers that behave exactly in this manner in retail outlets here in the past. Many timers do the count down part but not the reset and recycle so you'd need to be selective. 

Many digital watches have an alarm that can be set to sound at the same time every day. Some also have repeating timers. 


The alarm has the advantage that you can set it at any time to sound at say midnight but they can not be set for say 23 hour cycles. 
The timer has the flexibility of any length of cycle but you need to start it at the start of a cycle (eg you need to be awake at midnight if you want the cycles to start from midnight. 


Almost any microcontroller or an Arduino or whatever could be programmed for this task. Crystal control is desirable to allow reasonable time accuracy. 

(2) USB blipper.

Once ypu have a timer triggering you could possible access a DC signal which activates a sounder but can for certain access the sound signal either by direct connection or using a microphone or piezo sounder disc used as a pickup. Once you have an audio signal you can use it to activate a relay either for the duration of the tone or for a one off period at the start or end of the tine. This could be done with a transistor or few or a pkg of opamps (eg LM324for 

Any electronics including the timer could be powered by the USB +5V. The timer needs to be able to be run from +5V - some run from a 9V battery but will operate at 5V - this needs to be checked. Many operate from 1x AA or 2 x AA batteries and can easily be powered from 5V. Worst case you could make a 5V to 9V or whatever power supply but there should be enough timers available to allow a 5V capable one to be selected. 

The Relay ideally should have  4 pole normally closed contacts (4PSTNC) but as little as a 1 pole normally closed will usually work.  4PSTCNC allows you to breal all 4 USB wires cleanly. Next best is breaking just power leads which will quite possibly achieve your aim. A single contact in one power lead will also quite likely work. I suggest 4PSTNC as worst case the modem may have a path to ground via equipment connected to it and just opening the power pair or one power lead MAY not do what you want. Testing is easy. 

Once an adequate relay configuration is found, cut a USB extension cable in two (one or 2 or 4 conductors as require cut with the rest straight through) and route the cut wires via the relay contacts.

QED.