Can I limit the time a table lock is being held server side? I have a SQL Azure server with some stored procedures with TABLOCKX hints that I call from C# code on a Windows Azure web role VM. 

I'm kinda paranoid about those hints - what if there's some network protocol failure and the transaction doesn't end or something like that and a lock is held forever and all invokations of that stored procedure block on the statement with the TABLOCKX hint and all my stuff comes to a halt.

Is it somehow possible to tell SQL Server that the lock should not be held for more than say one minute so that it is unconditionally released after that time?
You can explicitly specify using :

SET LOCK_TIMEOUT timeout_period


  timeout_period
  
  Is the number of milliseconds that will pass before Microsoft SQL Server returns a locking error. A value of -1 (default) indicates no time-out period (that is, wait forever).
  When a wait for a lock exceeds the time-out value, an error is returned. A value of 0 means to not wait at all and return a message as soon as a lock is encountered.
  
  At the beginning of a connection, this setting has a value of -1. After it is changed, the new setting stays in effect for the remainder of the connection.
  
  The setting of SET LOCK_TIMEOUT is set at execute or run time and not at parse time.
  
  The READPAST locking hint provides an alternative to this SET option.
  
  CREATE DATABASE, ALTER DATABASE, and DROP DATABASE statements do not honor the SET LOCK_TIMEOUT setting.


Refer to : Support for SET statements in Azure

UPDATE: 


  Is it somehow possible to tell SQL Server that the lock should not be held for more than say one minute so that it is unconditionally released after that time?


No. As I mentioned in the comments section. 

THe database wont be able to maintain its ACID properties.

Either the query fails and returns an error or the lock is held by SQL Server  until the transaction is completed and then released.