I dont know whether you mean Alicante or Alicante Region,
however there are various options available,
There is reasonably good coverage of 3g in Alicante via vodaphone
U could go wifi by going near to one of the free outlets, which are invariably around, (for example in the village I live in they have provided a free Wifi, so you just take your laptop and sit outside the community centre and log on !!
Dial up via mobile, ( very slow and limited)
Satellite link Internet up and down, pricey Install about 2000? plus 65? per month (installed within a couple of days)
SKY DSL, dial up phone link upload, fast downlink via Sat dish and receiver, about 250? installed and 35? per month
If you have landline Telemonica (Pandemonia!!) with ADSL, if your lucky you may get connected in amonth or two
Or local provider via microlink,or landline if u have one, IE Comenersol or CitrusRED S.L. (you may have to search to find out who is local as for some strange reason they seem not to advertise well!!L