Flights
Use a comparison site for curent fares.KLM and Iberia and Turkish fly to Ataturk from Madrid
You can do Easyjet on a Sat. Alicante to Luton then Luton to other Istanbul Airport, but do not get in till 19.25. back to Alicante at 18.00. If one has to overnight may be better at Luton, and get into Istanbul at 15.30 next day. On balance LGW seems a better option.
Also Easyjet flight from Gatwick. In fact the cheapest way seems to be to flyovernight to Gatwick, then dawn flight to Istanbul. Coming back Istanbul 12.55 gets to LGW at 14.55 and can go on to Alicante (depends on day of week on which fight onwards)
By rental car :
The road from Istanbul to Eceabat is duel carriageway most of the way and the route is straighforward, so you should be able to get to the hotel without any delays or getting lost. You should be able to cover the distance of 320 km in 4 to 5 hours, allowing for a short break in Tekirdağ.
In general Turkish (highway) traffic is Mediterranean and might at first involve - for those not familiar with it- some white knuckle driving. Some golden rules : drive defensively, always exercising care. The Gallipoli Peninsula, on the other hand, is the perfect place to venture out with "your own wheels" : country roads with limited traffic.
Car rental about £200